Cartesian Product of Sets

<!DOCTYPE html>    <html lang="en-US" prefix="og: http://ogp.me/ns#" >  <head> <meta charset="UTF-8" /> <meta name="keywords" content="Data Structures,Algorithms,Python,Java,C,C++,JavaScript,Android Development,SQL,Data Science,Machine Learning,PHP,Web Development,System Design,Tutorial,Technical Blogs,Interview Experience,Interview Preparation,Programming,Competitive Programming,Jobs,Coding Contests,GATE CSE,HTML,CSS,React,NodeJS,Placement,Aptitude,Quiz,Computer Science,Programming Examples,GeeksforGeeks Courses,Puzzles,SSC,Banking,UPSC,Commerce,Finance,CBSE,School,k12,General Knowledge,News,Mathematics,Exams" /> <meta name="viewport" content="width=device-width, initial-scale=1.0, minimum-scale=0.5, maximum-scale=3.0"> <link rel="shortcut icon" href="https://media.geeksforgeeks.org/wp-content/cdn-uploads/gfg_favicon.png" type="image/x-icon" /> <link rel="preconnect" href="https://fonts.googleapis.com"> <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin> <meta name="theme-color" content="#308D46" /> <meta name='robots' content='index, follow, max-image-preview:large, max-snippet:-1' /> <meta name="image" property="og:image" content="https://media.geeksforgeeks.org/wp-content/cdn-uploads/gfg_200x200-min.png"> <meta property="og:image:type" content="image/png"> <meta property="og:image:width" content="200"> <meta property="og:image:height" content="200"> <meta name="facebook-domain-verification" content="xo7t4ve2wn3ywfkjdvwbrk01pvdond" /> <script src="https://cdnads.geeksforgeeks.org/synchronously_gfg_ads.min.js"></script> <script defer src="https://apis.google.com/js/platform.js"></script> <script src="//cdnjs.cloudflare.com/ajax/libs/require.js/2.1.14/require.min.js"></script>     <script> var IHPWT={}; //Initialize Namespace var pbjs = pbjs || {}; pbjs.que = pbjs.que || []; var googletag = googletag || {}; googletag.cmd = googletag.cmd || []; var gptRan = false; </script> <script defer src="https://ads.pubmatic.com/AdServer/js/pwt/162080/12331/pwt.js"></script> <script defer src="https://securepubads.g.doubleclick.net/tag/js/gpt.js"></script> <script defer src="https://cdnads.geeksforgeeks.org/prebid.js?ver=0.1"></script> <script defer src="https://cdnads.geeksforgeeks.org/gfg_ads.min.js?ver=0.1"></script>  <title>Cartesian Product of Sets - GeeksforGeeks</title> <link rel="profile" href="http://gmpg.org/xfn/11" /> <link rel="pingback" href="" />   <script> var video_api_response = JSON.parse("{\"id\":8965,\"title\":\"Cartesian Product of Sets\",\"slug\":\"cartesian-product-of-sets\",\"description\":\"<p style=\\\"margin-left:0px;\\\">Explore how to find the Cartesian product of sets with our comprehensive tutorial. This guide is perfect for computer science students, programmers, and anyone interested in understanding set operations and combinatorial techniques.<\/p><p style=\\\"margin-left:0px;\\\">In this tutorial, you'll learn:<\/p><p style=\\\"margin-left:0px;\\\">Understanding the Cartesian Product: Gain a foundational understanding of the Cartesian product, which involves creating ordered pairs from two sets. Each pair consists of one element from the first set and one element from the second set.<\/p><p style=\\\"margin-left:0px;\\\">Mathematical Definition: The Cartesian product of two sets \ud835\udc34<i>A<\/i> and \ud835\udc35<i>B<\/i>, denoted as \ud835\udc34\u00d7\ud835\udc35<i>A<\/i>\u00d7<i>B<\/i>, is defined as:<\/p><p>\ud835\udc34\u00d7\ud835\udc35={(\ud835\udc4e,\ud835\udc4f)\u2223\ud835\udc4e\u2208\ud835\udc34 and \ud835\udc4f\u2208\ud835\udc35}<i>A<\/i>\u00d7<i>B<\/i>={(<i>a<\/i>,<i>b<\/i>)\u2223<i>a<\/i>\u2208<i>A<\/i> and <i>b<\/i>\u2208<i>B<\/i>}<\/p><p style=\\\"margin-left:0px;\\\">This means that each element \ud835\udc4e<i>a<\/i> from set \ud835\udc34<i>A<\/i> is paired with every element \ud835\udc4f<i>b<\/i> from set \ud835\udc35<i>B<\/i>.<\/p><p style=\\\"margin-left:0px;\\\">Approach and Strategy:<\/p><ul><li><strong>Iterative Approach<\/strong>: Learn how to use nested loops to generate the Cartesian product of two sets. This straightforward method involves iterating through each element of the first set and pairing it with each element of the second set.<\/li><li><strong>Using List Comprehensions<\/strong>: Discover how to use list comprehensions in Python to generate the Cartesian product concisely and efficiently.<\/li><\/ul><p style=\\\"margin-left:0px;\\\">Algorithm Steps:<\/p><p style=\\\"margin-left:0px;\\\"><strong>Iterate Through Sets<\/strong>:<\/p><ul><li>Use nested loops to iterate through each element in the first set and each element in the second set.<\/li><li>Create pairs consisting of one element from the first set and one element from the second set.<\/li><li>Store these pairs in a list or set.<\/li><\/ul><p style=\\\"margin-left:0px;\\\"><strong>List Comprehensions<\/strong> (Python-specific):<\/p><ul><li>Use a single line of code with list comprehensions to generate the Cartesian product.<\/li><\/ul><p style=\\\"margin-left:0px;\\\">Step-by-Step Code Implementation: Detailed code examples in popular programming languages like Python, Java, and C++. These examples will demonstrate both the iterative approach and the use of list comprehensions to generate the Cartesian product efficiently.<\/p><p style=\\\"margin-left:0px;\\\">Complexity Analysis: Discuss the time and space complexity of the solution. The time complexity is \ud835\udc42(\ud835\udc5a\u00d7\ud835\udc5b)<i>O<\/i>(<i>m<\/i>\u00d7<i>n<\/i>), where \ud835\udc5a<i>m<\/i> is the number of elements in the first set and \ud835\udc5b<i>n<\/i> is the number of elements in the second set. The space complexity is also \ud835\udc42(\ud835\udc5a\u00d7\ud835\udc5b)<i>O<\/i>(<i>m<\/i>\u00d7<i>n<\/i>) for storing the resulting pairs.<\/p><p style=\\\"margin-left:0px;\\\">Handling Edge Cases: Tips on managing various edge cases such as:<\/p><ul><li><strong>Empty Sets<\/strong>: Ensure the implementation handles cases where one or both sets are empty correctly, resulting in an empty Cartesian product.<\/li><li><strong>Sets with One Element<\/strong>: Consider how the solution works when one or both sets contain only one element.<\/li><\/ul><p style=\\\"margin-left:0px;\\\">Visual Examples and Diagrams: Include diagrams to visually demonstrate the process of creating ordered pairs and forming the Cartesian product, helping to clarify the steps involved.<\/p><p style=\\\"margin-left:0px;\\\">Applications and Real-World Use: Discuss real-world applications of the Cartesian product, such as in database queries (cross joins), generating combinations for testing, and forming grids or matrices in computational problems.<\/p><p style=\\\"margin-left:0px;\\\">By the end of this tutorial, you\u2019ll be well-equipped to generate the Cartesian product of two sets using both iterative and concise methods, enhancing your problem-solving skills and your understanding of set operations.<\/p><p style=\\\"margin-left:0px;\\\">For a comprehensive guide on finding the Cartesian product of sets, including detailed explanations, code examples, and practical tips, check out our full article at <a href=\\\"https:\/\/www.geeksforgeeks.org\/cartesian-product-of-sets\/\\\">https:\/\/www.geeksforgeeks.org\/cartesian-product-of-sets\/<\/a>.<\/p><p style=\\\"margin-left:0px;\\\">This tutorial will not only improve your understanding of set operations and combinatorial techniques but also prepare you to tackle similar challenges in your software development and algorithmic problem-solving endeavors.<\/p>\",\"source\":\"https:\/\/videos.geeksforgeeks.org\/hls\/415ee5418b66d62de8d1a2008bf70440gfg-CartesianProductofSets20240517124333.m3u8\",\"category\":[{\"term_id__id\":15,\"term_id__term_name\":\"Maths\",\"term_id__term_type\":1,\"term_id__slug\":\"maths\"},{\"term_id__id\":16,\"term_id__term_name\":\"Maths\",\"term_id__term_type\":2,\"term_id__slug\":\"maths-kwitto\"}],\"video_type\":\"geeksforgeeks\",\"meta\":{\"thumbnail\":\"https:\/\/media.geeksforgeeks.org\/geeksforgeeks\/CartesianProductofSets\/CartesianProductofSets20240517124338-small.png\",\"largeThumbnail\":\"https:\/\/media.geeksforgeeks.org\/geeksforgeeks\/CartesianProductofSets\/CartesianProductofSets20240517124338.jpg\",\"likes\":0,\"views\":500,\"isFeatured\":0,\"isPremium\":0,\"isPublic\":0,\"format\":\"video\/mp4\",\"revision\":{}},\"time\":\"17\/05\/2024\",\"seo\":{\"description\":\"<p style=\\\"margin-left:0px;\\\">Explore how to find the Cartesian product of se\",\"thumbnail\":\"https:\/\/media.geeksforgeeks.org\/geeksforgeeks\/CartesianProductofSets\/CartesianProductofSets20240517124338-seo.png\"},\"subtitle\":\"https:\/\/videos.geeksforgeeks.org\/subtitles\/CartesianProductofSets20240517124333.vtt\",\"duration\":505,\"post_list\":[{\"post_id\":504094,\"post_title\":\"Cartesian Product of Sets\"}],\"course_link\":null,\"views\":\"500\",\"video_schema\":{\"@context\":\"https:\/\/schema.org\",\"@type\":\"VideoObject\",\"name\":\"Cartesian Product of Sets\",\"description\":\"Explore how to find the Cartesian product of sets with our comprehensive tutorial. This guide is perfect for computer science students, programmers, and anyone interested in understanding set operations and combinatorial techniques.In this tutorial, youll learnUnderstanding the Cartesian Product Gain a foundational understanding of the Cartesian product, which involves creating ordered pairs from two sets. Each pair consists of one element from the first set and one element from the second set.Mathematical Definition The Cartesian product of two sets A and B, denoted as AB, is defined as(,) nbspand nbspAB(a,b)aA nbspand nbspbBThis means that each element a from set A is paired with every element b from set B.Approach and StrategyIterative Approach Learn how to use nested loops to generate the Cartesian product of two sets. This straightforward method involves iterating through each element of the first set and pairing it with each element of the second set.Using List Comprehensions Discover how to use list comprehensions in Python to generate the Cartesian product concisely and efficiently.Algorithm StepsIterate Through SetsUse nested loops to iterate through each element in the first set and each element in the second set.Create pairs consisting of one element from the first set and one element from the second set.Store these pairs in a list or set.List Comprehensions (Python-specific)Use a single line of code with list comprehensions to generate the Cartesian product.Step-by-Step Code Implementation Detailed code examples in popular programming languages like Python, Java, and C. These examples will demonstrate both the iterative approach and the use of list comprehensions to generate the Cartesian product efficiently.Complexity Analysis Discuss the time and space complexity of the solution. The time complexity is ()O(mn), where m is the number of elements in the first set and n is the number of elements in the second set. The space complexity is also ()O(mn) for storing the resulting pairs.Handling Edge Cases Tips on managing various edge cases such asEmpty Sets Ensure the implementation handles cases where one or both sets are empty correctly, resulting in an empty Cartesian product.Sets with One Element Consider how the solution works when one or both sets contain only one element.Visual Examples and Diagrams Include diagrams to visually demonstrate the process of creating ordered pairs and forming the Cartesian product, helping to clarify the steps involved.Applications and Real-World Use Discuss real-world applications of the Cartesian product, such as in database queries (cross joins), generating combinations for testing, and forming grids or matrices in computational problems.By the end of this tutorial, youll be well-equipped to generate the Cartesian product of two sets using both iterative and concise methods, enhancing your problem-solving skills and your understanding of set operations.For a comprehensive guide on finding the Cartesian product of sets, including detailed explanations, code examples, and practical tips, check out our full article at httpswww.geeksforgeeks.orgcartesian-product-of-sets.This tutorial will not only improve your understanding of set operations and combinatorial techniques but also prepare you to tackle similar challenges in your software development and algorithmic problem-solving endeavors.\",\"thumbnailUrl\":[\"https:\/\/media.geeksforgeeks.org\/geeksforgeeks\/CartesianProductofSets\/CartesianProductofSets20240517124338-small.png\",\"https:\/\/media.geeksforgeeks.org\/geeksforgeeks\/CartesianProductofSets\/CartesianProductofSets20240517124338-seo.png\",\"https:\/\/media.geeksforgeeks.org\/geeksforgeeks\/CartesianProductofSets\/CartesianProductofSets20240517124338.jpg\"],\"uploadDate\":\"2024-05-17T12:44:58Z\",\"duration\":\"PT0H8M25S\",\"contentUrl\":\"https:\/\/www.geeksforgeeks.org\/videos\/cartesian-product-of-sets\/\"}}"); </script> <script type="application/ld+json">{"@context":"https://schema.org","@type":"VideoObject","name":"Cartesian Product of Sets","description":"Explore how to find the Cartesian product of sets with our comprehensive tutorial. 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This straightforward method involves iterating through each element of the first set and pairing it with each element of the second set.Using List Comprehensions Discover how to use list comprehensions in Python to generate the Cartesian product concisely and efficiently.Algorithm StepsIterate Through SetsUse nested loops to iterate through each element in the first set and each element in the second set.Create pairs consisting of one element from the first set and one element from the second set.Store these pairs in a list or set.List Comprehensions (Python-specific)Use a single line of code with list comprehensions to generate the Cartesian product.Step-by-Step Code Implementation Detailed code examples in popular programming languages like Python, Java, and C. These examples will demonstrate both the iterative approach and the use of list comprehensions to generate the Cartesian product efficiently.Complexity Analysis Discuss the time and space complexity of the solution. The time complexity is ()O(mn), where m is the number of elements in the first set and n is the number of elements in the second set. 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For example, 45 is the product of 9 and 5. One must be familiar with the basic operations on sets like Union and Intersection, which are performed on 2 or more sets. 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This means (a, b) is distinct from (b, a) if a is not equal to b." } }, { "@type": "Question", "name": "What is cartesian product of 3 sets?", "acceptedAnswer": { "@type": "Answer", "text": "Cartesian product of three sets A, B, and C is the set of all possible ordered triples where first element is from A, second from B, and third from C." } }, { "@type": "Question", "name": "Write formula for cartesian product of sets.", "acceptedAnswer": { "@type": "Answer", "text": "Cartesian product of two sets A and B is defined as:A × B = {(a, b) | a ∈ A and b ∈ B}For three sets A, B, and C:A × B × C = {(a, b, c) | a ∈ A, b ∈ B, and c ∈ C}" } }, { "@type": "Question", "name": "What is cartesian product of a set and a null set?", "acceptedAnswer": { "@type": "Answer", "text": "Cartesian product of a set A and a null set (∅) is always an empty set (∅), as there are no elements in the null set to form pairs with elements from set A." } } ] }</script> <script> var arrPostCat = new Array(); var arrPostCatName="Mathematics"; arrPostCat.push('4684'); arrPostCat.push('4667'); arrPostCat.push('4677'); var tIds = "4684,4667,4677,5966,2963"; var termsNames = "Class11,SchoolLearning,Mathematics,MathsClass11,Picked"; var tIdsInclusiveParents = "4684,4667,4677,5966,2963" var domain = 1; var arrPost = new Array(); var post_id = "504094"; var post_type = "post"; var post_slug = window.location.href; var ip = "64.252.72.18"; var post_title = `Cartesian Product of Sets`; var post_status = "publish"; var practiceAPIURL="https://practiceapi.geeksforgeeks.org/"; var practiceURL="https://practice.geeksforgeeks.org/"; var post_date = "2020-11-16 00:47:33"; var commentSysUrl = "https://discuss.geeksforgeeks.org/commentEmbedV2.js"; //var postAdApiUrlString = ""; var link_on_code_run = ''; var link_search_modal_top = ''; var country_code_cf = "DE"; var postAdApiUrlString = '4677/4684/4667/4677/5966/2963/'; </script>  <meta name="description" content="A Computer Science portal for geeks. 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target="_self">Basic</a></li><li class="mega-dropdown__list-item" data-child="true" aria-expanded="false" data-expandable="false"><a href="https://www.geeksforgeeks.org/explore?page=1&difficulty=Easy&itm_source=geeksforgeeks&itm_medium=main_header_outIndia&itm_campaign=DSA_Header" target="_self">Easy</a></li><li class="mega-dropdown__list-item" data-child="true" aria-expanded="false" data-expandable="false"><a href="https://www.geeksforgeeks.org/explore?page=1&difficulty=Medium&itm_source=geeksforgeeks&itm_medium=main_header_outIndia&itm_campaign=DSA_Header" target="_self">Medium</a></li><li class="mega-dropdown__list-item" data-child="true" aria-expanded="false" data-expandable="false"><a href="https://www.geeksforgeeks.org/explore?page=1&difficulty=Hard&itm_source=geeksforgeeks&itm_medium=main_header_outIndia&itm_campaign=DSA_Header" target="_self">Hard</a></li></ul></li><li class="mega-dropdown__list-item" data-parent="false" aria-expanded="true" 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data-child="true" aria-expanded="false" data-expandable="false"><a href="https://www.geeksforgeeks.org/explore?page=1&sprint=57184072610b884e5df3584cc534115d&sortBy=submissions&sprint_name=Top 50 String Problems&itm_source=geeksforgeeks&itm_medium=main_header_outIndia&itm_campaign=DSA_Header" target="_self">Top 50 String Problems</a></li><li class="mega-dropdown__list-item" data-child="true" aria-expanded="false" data-expandable="false"><a href="https://www.geeksforgeeks.org/explore?page=1&sprint=93d672753b74440c7427214c8ebf866d&sortBy=submissions&sprint_name=Top 50 DP Problems&itm_source=geeksforgeeks&itm_medium=main_header_outIndia&itm_campaign=DSA_Header" target="_self">Top 50 DP Problems</a></li><li class="mega-dropdown__list-item" data-child="true" aria-expanded="false" data-expandable="false"><a href="https://www.geeksforgeeks.org/explore?page=1&sprint=405e9db0f353691ad3b2d546b19145e9&sortBy=submissions&sprint_name=Top 50 Graph Problems&itm_source=geeksforgeeks&itm_medium=main_header_outIndia&itm_campaign=DSA_Header" target="_self">Top 50 Graph Problems</a></li><li class="mega-dropdown__list-item" data-child="true" aria-expanded="false" data-expandable="false"><a href="https://www.geeksforgeeks.org/explore?page=1&sprint=5f24de84b65bf7c4f4399c8111e26b81&sortBy=submissions&sprint_name=Top 50 Tree Problems&itm_source=geeksforgeeks&itm_medium=main_header_outIndia&itm_campaign=DSA_Header" target="_self">Top 50 Tree Problems</a></li></ul></li></ul></li><li class="mega-dropdown__list-item" data-child="true" aria-expanded="false" data-expandable="false"><a href="https://www.geeksforgeeks.org/competitive-programming-a-complete-guide/?ref=outind" target="_self">Competitive Programming</a></li><li class="mega-dropdown__list-item" data-parent="false" aria-expanded="true" data-expandable="true"><span>Company Wise SDE Sheets</span><i class="gfg-icon gfg-icon_arrow-right"></i><ul class="mega-dropdown Screen_2"><li 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aria-expanded="false" data-expandable="false"><a href="https://www.geeksforgeeks.org/must-do-coding-questions-for-companies-like-amazon-microsoft-adobe/?ref=outind" target="_self">FAANG Coding Sheet</a></li><li class="mega-dropdown__list-item" data-child="true" aria-expanded="false" data-expandable="false"><a href="https://www.geeksforgeeks.org/must-do-coding-questions-for-product-based-companies/?ref=outind" target="_self">Product-Based Coding Sheet</a></li><li class="mega-dropdown__list-item" data-child="true" aria-expanded="false" data-expandable="false"><a href="https://www.geeksforgeeks.org/must-coding-questions-company-wise/?ref=outind" target="_self">Company-Wise Preparation Sheet</a></li></ul></li><li class="mega-dropdown__list-item" data-child="true" aria-expanded="false" data-expandable="false"><a href="https://www.geeksforgeeks.org/top-100-data-structure-and-algorithms-dsa-interview-questions-topic-wise/?ref=outind" target="_self">Top Interview Questions</a></li><li 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data-parent="false" aria-expanded="true" data-expandable="true"><span>Data Science</span><i class="gfg-icon gfg-icon_arrow-down gfg-icon_header"></i><ul class="mega-dropdown Screen_1"><li class="mega-dropdown__list-item" data-child="true" aria-expanded="false" data-expandable="false"><a href="https://www.geeksforgeeks.org/python-programming-language/?ref=outind" target="_self">Python Tutorial</a></li><li class="mega-dropdown__list-item" data-child="true" aria-expanded="false" data-expandable="false"><a href="https://www.geeksforgeeks.org/r-tutorial/?ref=outind" target="_self">R Tutorial</a></li><li class="mega-dropdown__list-item" data-child="true" aria-expanded="false" data-expandable="false"><a href="https://www.geeksforgeeks.org/machine-learning/?ref=outind" target="_self">Machine Learning</a></li><li class="mega-dropdown__list-item" data-child="true" aria-expanded="false" data-expandable="false"><a href="https://www.geeksforgeeks.org/data-science-with-python-tutorial/?ref=outind" 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</div> </div> </div> <div class="text"> <p dir="ltr"><span>‘</span><b><strong>Product</strong></b><span>‘ mathematically signifies the result obtained when two or more values are multiplied together. For example, 45 is the product of 9 and 5. One must be familiar with the basic operations on sets like Union and Intersection, which are performed on 2 or more sets. </span></p> <p dir="ltr"><span>Cartesian Product is also one such operation that is performed on two sets, which</span><b><strong> returns a set of ordered pairs</strong></b><span>.</span></p><div id="GFG_AD_gfg_mobile_336x280" style="margin: 5px 0;"></div> <p dir="ltr"><span>In this article, we have covered, the ordered pair definition, a cartesian product of sets, and others in detail.</span></p> <div id="table_of_content"> <p style="margin:4px; font-size:20px; font-weight:bold;">Table of Content</p> <ul> <li><a href="#what-is-an-ordered-pair">What is an Ordered Pair?</a></li> <li><a href="#cartesian-product-of-sets">Cartesian Product of Sets</a></li> <li><a href="#properties-of-cartesian-product">Properties of Cartesian Product</a></li> <li><a href="#problems-on-cartesian-product-of-sets">Problems on Cartesian Product of Sets</a></li> </ul> </div> <h2 id="what-is-an-ordered-pair"><b><strong>What is an Ordered Pair?</strong></b></h2> <p dir="ltr"><span>An ordered pair has two parts. The first part is called the first component. The second part is called the second component. We write an ordered pair like this: (a, b). The letter ‘a’ is the first component. The letter ‘b’ is the second component. An ordered pair has two things. One thing comes first. The other thing comes second.</span></p> <p dir="ltr"><b><strong>Example:</strong></b></p> <blockquote> <p dir="ltr"><span>(5, 7) is an ordered pair of integers.</span></p> </blockquote><div id="GFG_AD_Desktop_InContent_ATF_336x280" style="text-align:center; max-height: 280px;"></div><div id="GFG_AD_gfg_mobile_336x280_1" style="margin: 5px 0;"></div> <p dir="ltr"><b><strong>Note:</strong></b><span> (5, 7) ≠ (7, 5), an ordered pair (a, b) is equal to (x, y) only if a = x and b = y.</span></p><div id="_GFG_ABP_Incontent_728x90" style="text-align:center;"></div> <h2 id="cartesian-product-of-sets"><b><strong>Cartesian Product of Sets</strong></b></h2> <p dir="ltr"><span>When two sets have items in them, A and B, their Cartesian product is all the pairs you can make. One part of the pair comes from set A. The other part comes from set B. We make every possible pair this way. The result is a new set of all these pairs. We write this new set as A×B.</span></p> <p dir="ltr"><b><strong>A × B = {(a, b) : a ∈ A and b ∈ B}</strong></b></p> <p dir="ltr"><b><strong>Example: </strong></b></p> <blockquote> <p dir="ltr"><span>Let A = {1, 2} and B = {4, 5, 6}</span></p> <p dir="ltr"><span>A × B = {(1, 4), (1, 5), (1, 6), (2, 4), (2, 5), (2, 6)}</span></p><div style="text-align:center;margin:20px 0; max-height: 280px;" id="GFG_AD_gfg_outstream_incontent"></div> </blockquote> <p dir="ltr"><span>Here the first component of every ordered pair is from set A the second component is from set B.</span></p> <p dir="ltr"><span>Cartesian Product of two sets can be easily represented in the form of a matrix where both sets are on either axis, as shown in the image below. Cartesian Product of  A = {1,  2} and B = {x, y, z}</span></p> <p dir="ltr"><img src="https://media.geeksforgeeks.org/wp-content/uploads/20201015194740/CartesianProduct.png" alt="Cartesian Product of two sets" width="inherit" height="inherit"></p> <h2 id="properties-of-cartesian-product"><b><strong>Properties of Cartesian Product</strong></b></h2> <p dir="ltr"><span>Various properties of cartesian product includes,</span></p> <h3 id="1-cartesian-product-is-noncommutative-a-b-b-a"><b><strong>1. Cartesian Product is non-commutative: A × B ≠ B × A</strong></b></h3> <p dir="ltr"><b><strong>Example: </strong></b></p> <blockquote> <p dir="ltr"><span>A = {1, 2} , B = {a, b}</span></p> <p dir="ltr"><span>A × B = {(1, a), (1, b), (2, a), (2, b)}</span></p> <p dir="ltr"><span>B × A = {(a, 1), (b, 1), (b, 1), (b, 2)}</span></p> <p dir="ltr"><span>Therefore as A ≠ B we have A × B ≠ B × A</span></p> </blockquote> <h3 id="2-a-b-b-a-only-if-a-b"><b><strong>2.</strong></b><span> </span><b><strong>A × B = B × A, only if A = B</strong></b></h3> <p dir="ltr"><b><strong>Proof:</strong></b></p><div style="text-align:center;margin:20px 0;max-height:280px" id="GFG_AD_gfg_outstream_incontent-1"></div> <blockquote> <p dir="ltr"><span>Let A × B = B × A then we have  </span></p> <p dir="ltr"><span>A ⊆  B  and B ⊆  A, it follows that A = B</span></p> </blockquote> <h3 id="3-cardinality-of-cartesian-product-is-defined-as-number-of-elements-in-a-b-and-is-equal-to-the-product-of-cardinality-of-both-sets-ie"><b><strong>3. Cardinality of Cartesian Product is defined as number of elements in A × B and is equal to the product of cardinality of both sets i.e.,</strong></b></h3> <h3 id="a-b-a-b"><b><strong>|A × B| = |A| × |B|</strong></b></h3> <p dir="ltr"><b><strong>Proof:</strong></b></p> <blockquote> <p dir="ltr"><span>Let a ∈ A then the number of ordered pair (a, b) such that b ∈ B is |B|</span></p> <p dir="ltr"><span>Therefore we have |B| choices for b for each a where a ∈ A therefore the number of element in A × B is |A| × |B|</span></p> </blockquote> <p dir="ltr"><b><strong>4. A × B = ∅, if either A = ∅ or B = ∅</strong></b></p> <p dir="ltr"><b><strong>Proof:</strong></b><span> </span></p> <blockquote> <p dir="ltr"><span>Suppose A×B=∅. This means there are no ordered pairs (a,b) where a∈A and b∈B.</span></p> <p dir="ltr"><span>If A is non-empty, then there exists at least one element a∈A. For any such a, there should be an ordered pair (a,b) for some b∈B, as B is not empty. But since we have assumed A×B=∅, this is a contradiction. Hence, A must be empty.</span></p> <p dir="ltr"><span>Similarly, if B is non-empty, then there exists at least one element b∈B. For any such b, there should be an ordered pair (a,b) for some a∈A, as A is not empty. But since we have assumed A×B=∅, this is a contradiction. Hence, B must be empty.</span></p><div style="text-align:center;margin:20px 0;max-height:280px" id="GFG_AD_gfg_outstream_incontent-2"></div> <p dir="ltr"><span>Therefore, if ? × ? = ∅, either A or B must be empty</span></p> <p dir="ltr"><span>Hence, the statement ? × ? = ∅ if and only if either A=∅ or ? = ∅ is proven.</span></p> </blockquote> <p dir="ltr"><b><strong>Read More,</strong></b></p> <blockquote> <ul> <li value="1"><a href="https://www.geeksforgeeks.org/set-theory/?ref=header_search" rel="noopener"><b><strong>Set Theory</strong></b></a></li> <li value="2"><a href="https://www.geeksforgeeks.org/ordered-pair/?ref=header_search" rel="noopener"><b><strong>Ordered Pair</strong></b></a></li> <li value="3"><a href="https://www.geeksforgeeks.org/finite-sets/?ref=header_search" rel="noopener"><b><strong>Finite Set</strong></b></a></li> </ul> </blockquote> <h2 id="problems-on-cartesian-product-of-sets"><b><strong>Problems on Cartesian Product of Sets</strong></b></h2> <p dir="ltr"><b><strong>Problem 1: Find the value of x and y given (2x – y,  25) = (15,  2x + y)?</strong></b></p> <p dir="ltr"><b><strong>Solution:  </strong></b></p> <blockquote> <p dir="ltr"><span>As we know from the property of ordered pairs, 2x – y = 15 and 25 = 2x + y.</span></p> <p dir="ltr"><span>Solving the linear equations we have x = 10 and y = 5.</span></p> </blockquote> <p dir="ltr"><b><strong>Problem 2. Given A = {2, 3, 4 , 5} and B = {4 , 16 , 23}, a ∈ A, b ∈ B, find the set of ordered pairs such that a</strong></b><b><sup><strong>2</strong></sup></b><b><strong> < b?</strong></b></p> <p dir="ltr"><b><strong>Solution:</strong></b></p> <blockquote> <p dir="ltr"><span>As 2</span><sup><span>2</span></sup><span> < 16 and 23, 3</span><sup><span>2</span></sup><span> < 16 and 23, 4</span><sup><span>2</span></sup><span> < 23  </span></p><div style="text-align:center;margin:20px 0;max-height:280px" id="GFG_AD_gfg_outstream_incontent-3"></div> <p dir="ltr"><span>We have the set of ordered pairs such that a</span><sup><span>2</span></sup><span> < b is {(2, 16), (2, 23), (3, 16), (2, 23), (4, 23)}</span></p> </blockquote> <p dir="ltr"><b><strong>Problem 3. If A = {9, 10} and B = {3, 4, 6}, find A × B and |A × B|? </strong></b></p> <p dir="ltr"><i><b><strong class="GFGEditorTheme__textItalic">S</strong></b></i><b><strong>olution:</strong></b></p> <blockquote> <p dir="ltr"><span>A × B = {(9, 3), (9, 4), (9, 6), (10, 3), (10, 4), (10, 6)}</span></p> <p dir="ltr"><span>|A × B| = |A| * |B| = 2 * 3 = 6</span></p> </blockquote> <p dir="ltr"><b><strong>Problem 4. If A × B = {(a, x), (a, y ), (b, x ), (b, y)}, find A and B?</strong></b></p> <p dir="ltr"><b><strong>Solution:</strong></b></p> <blockquote> <p dir="ltr"><span>We know A is the set of all first components in ordered pairs of A × B and </span></p> <p dir="ltr"><span>B is the set of the second component in the ordered pair of A × B.</span></p> <p dir="ltr"><span>Therefore A = {a, b} and B = {x, y}</span></p><div style="text-align:center;margin:20px 0;max-height:280px" id="GFG_AD_gfg_outstream_incontent-4"></div> </blockquote> <p dir="ltr"><b><strong>Problem 5. Given A × B has 15 ordered pairs and A has 5 elements, find the number of elements in B?</strong></b></p> <p dir="ltr"><b><strong>Solution:</strong></b></p> <blockquote> <p dir="ltr"><span>We know |A × B| = |A| * |B|, 15 = 5 * |B|</span></p> <p dir="ltr"><span>Therefore B has 15 / 5 = 3 elements.</span></p> </blockquote> <h2 id="conclusion"><span>Conclusion</span></h2> <p dir="ltr"><span>The Cartesian Product of Sets is a fundamental concept in set theory and mathematics that helps in understanding the combination of elements from the two or more sets. By creating ordered pairs from the elements of the sets it provides a structured way to explore relationships and combinations. The practice problems presented above illustrate the various scenarios where the Cartesian Product can be applied ranging from the simple sets to more complex combinations.</span></p> <h2 id="faqs-on-cartesian-product-of-sets"><span>FAQs on Cartesian Product of Sets</span></h2> <h3 id="what-is-cartesian-product-of-two-sets-ie-a-b"><span>What is cartesian product of two sets i.e. A </span><b><strong>× </strong></b><span>B?</span></h3> <blockquote> <p dir="ltr"><span>Cartesian product of sets A and B, denoted A×B, is the set of all possible ordered pairs where the first element is from A and the second from B.</span></p> </blockquote> <h3 id="define-ordered-pair"><span>Define ordered pair.</span></h3> <blockquote> <p dir="ltr"><span>An ordered pair is a pair of elements (a, b) in which the order of the elements is significant. This means (a, b) is distinct from (b, a) if a is not equal to b.</span></p> </blockquote> <h3 id="what-is-cartesian-product-of-3-sets"><span>What is cartesian product of 3 sets?</span></h3> <blockquote> <p dir="ltr"><span>Cartesian product of three sets A, B, and C is the set of all possible ordered triples where first element is from A, second from B, and third from C.</span></p> </blockquote> <h3 id="write-formula-for-cartesian-product-of-sets"><span>Write formula for cartesian product of sets.</span></h3> <blockquote> <p dir="ltr" style="text-align: start;"><span>Cartesian product of two sets A and B is defined as:</span></p> <p dir="ltr" style="text-align: center;"><b><strong>A × B = {(a, b) | a ∈ A and b ∈ B}</strong></b></p> <p dir="ltr" style="text-align: start;"><span>For three sets A, B, and C:</span></p> <p dir="ltr" style="text-align: center;"><b><strong>A × B × C = {(a, b, c) | a ∈ A, b ∈ B, and c ∈ C}</strong></b></p> </blockquote> <h3 id="what-is-cartesian-product-of-a-set-and-a-null-set"><span>What is cartesian product of a set and a null set?</span></h3> <blockquote> <p dir="ltr"><span>Cartesian product of a set A and a null set (∅) is always an empty set (∅), as there are no elements in the null set to form pairs with elements from set A.</span></p> </blockquote> <br/><div id="AP_G4GR_6"></div> <div class='article_bottom_text'></div><br> <div id="video-tab-content" class="video-tab-content"> <div style="text-align: center; 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Relation and Function in maths are analogous to the relation that we see in our daily lives i.e., two persons are related by the relation of father-son, mother-daughter, brother-sister, and many more. On a simi</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">13 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/introduction-to-domain-and-range-relations-and-functions/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Introduction to Domain and Range </div> <div class="gfg-similar-read-item-subheading">To understand the concept of Domain and range of a Relation first, we have to learn about Relation. A relation is a set of rules which relates the value of one set to the value of other sets. The domain of a relation is the set of values that we take as input and the range is the set of the values</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">7 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/piecewise-function/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Piecewise Function </div> <div class="gfg-similar-read-item-subheading">Piecewise Function is a function that behaves differently for two types of input. As we know a function is a mathematical object which associates each input with exactly one output. For example: If a function takes on any input and gives the output as 3. It can be represented mathematically as f(x)</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">11 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/range-of-a-function/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Range of a Function </div> <div class="gfg-similar-read-item-subheading">Functions in math can be thought of as vending machines. Given the money in the form of input, they give some cans or cookies in return. Similarly, functions take some input numbers and give us some output. It can be said that, in real life, Everything can be formulated and solved with the help of f</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">7 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li></ul></div><div class="leftbar-dropdown"><h2 class="dropdown-title">Chapter 3: Trigonometric Functions<span class="gfg-icon gfg-icon_arrow-down-thin"></span></h2><ul style="display: none;"><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/measuring-angles/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Measuring Angles </div> <div class="gfg-similar-read-item-subheading">Measurement of angles is very important in geometry for solving various geometrical problems and comparing angles in various geometrical figures. We use various geometric tools such as a protractor, and a compass for measuring angles. There are various types of angles measured by us and before learn</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">7 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/trigonometric-functions/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Trigonometric Functions </div> <div class="gfg-similar-read-item-subheading">Trigonometric Functions, often simply called trig functions, are mathematical functions that relate the angles of a right triangle to the ratios of the lengths of its sides. These Trigonometric functions are fundamental in mathematics for units like Calculus and Geometry and have wide-ranging applic</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">11 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/trigonometric-functions-of-sum-and-difference-of-two-angles/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Trigonometric Functions of Sum and Difference of Two Angles </div> <div class="gfg-similar-read-item-subheading">Trigonometry is a branch of mathematics, which deals with the angles, lengths, and heights of triangles and their relationships. It had played an important role in calculating complex functions or large distances which were not possible to calculate without trigonometry. While solving problems with</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">15+ min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li></ul></div><div class="leftbar-dropdown"><h2 class="dropdown-title">Chapter 4: Principle of Mathematical Induction<span class="gfg-icon gfg-icon_arrow-down-thin"></span></h2><ul style="display: none;"><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/principle-of-mathematical-induction/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Principle of Mathematical Induction </div> <div class="gfg-similar-read-item-subheading">Mathematical induction is a concept in mathematics that is used to prove various mathematical statements and theorems. The principle of mathematical induction is sometimes referred to as PMI. It is a technique that is used to prove the basic theorems in mathematics which involve the solution up to n</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">11 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li></ul></div><div class="leftbar-dropdown"><h2 class="dropdown-title">Chapter 5: Complex Numbers and Quadratic Equations<span class="gfg-icon gfg-icon_arrow-down-thin"></span></h2><ul style="display: none;"><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/complex-numbers/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Complex Numbers </div> <div class="gfg-similar-read-item-subheading">Complex numbers are an essential concept in mathematics, extending the idea of numbers to include solutions for equations that don't have real solutions. A complex number is expressed in the form a + bi, where a represents the real part, b represents the imaginary part, and i is the imaginary unit d</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">15+ min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/algebra-of-real-functions/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Algebra of Real Functions </div> <div class="gfg-similar-read-item-subheading">The algebra of real functions refers to the study and application of algebraic operations on functions that map real numbers to real numbers. A function can be thought of as a rule or set of rules which map an input to an output knows as its image. It is represented as x ⇢ Function ⇢ y. A real funct</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">5 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/algebraic-operations-on-complex-numbers-class-11-maths/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Algebraic Operations on Complex Numbers | Class 11 Maths </div> <div class="gfg-similar-read-item-subheading">A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i represents the imaginary unit, satisfying the equation i² = −1. For example, 5+6i is a complex number, where 5 is a real number and 6i is an imaginary number. Therefore, the combination of bo</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">9 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/polar-representation-of-complex-numbers/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Polar Representation of Complex Numbers </div> <div class="gfg-similar-read-item-subheading">Complex numbers, which take the form z = x + yi, can also be represented in a way that highlights their geometric properties. This alternative representation is known as the polar form. The polar representation of a complex number expresses it in terms of its magnitude (modulus) and direction (argum</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">10 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/absolute-value-of-a-complex-number/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Absolute Value of a Complex Number </div> <div class="gfg-similar-read-item-subheading">The absolute value, or modulus, of a complex number measures its distance from the origin on the complex plane. If you have a complex number z = a + ib, where a is the real part while ib is the imaginary part of the complex number in which i is known as iota and b is a real number. Then the absolute</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">7 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/complex-conjugate/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Conjugate of Complex Numbers </div> <div class="gfg-similar-read-item-subheading">Conjugate of Complex Number is a complex number obtained by changing the sign of the imaginary part. In simple words, conjugate of a complex number is a number that has the same real part as the original complex number, and the imaginary part has the same magnitude but opposite sign. In the world of</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">6 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/imaginary-numbers/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Imaginary Numbers </div> <div class="gfg-similar-read-item-subheading">Imaginary numbers are numbers as the name suggests are the number that is not real numbers. All the numbers real and imaginary come under the categories of complex numbers. Imaginary numbers are very useful in solving quadratic equations and other equations whose solutions can not easily be found us</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">9 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li></ul></div><div class="leftbar-dropdown"><h2 class="dropdown-title">Chapter 6: Linear Inequalities<span class="gfg-icon gfg-icon_arrow-down-thin"></span></h2><ul style="display: none;"><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/compound-inequality/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Compound Inequalities </div> <div class="gfg-similar-read-item-subheading">Compound Inequalities are the combination of two or more inequalities. These inequalities are combined using two conditions that are AND, and OR. These conditions have specific meanings and they are solved differently. The inequities in compound inequalities are individually solved using normal rule</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">11 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/algebraic-solutions-of-linear-inequalities-in-one-variable-and-their-graphical-representation-linear-inequalities-class-11-maths/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Algebraic Solutions of Linear Inequalities in One Variable and their Graphical Representation - Linear Inequalities | Class 11 Maths </div> <div class="gfg-similar-read-item-subheading">A linear inequality is a mathematical expression involving an inequality symbol (<, >, ≤, or ≥) and a linear expression. Unlike linear equations, which give a specific solution, linear inequalities define a range of possible solutions. Example: 2x+3>5 In this case, the inequality indicates</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">7 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/graphical-solution-of-linear-inequalities-in-two-variables/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Graphical Solution of Linear Inequalities in Two Variables </div> <div class="gfg-similar-read-item-subheading">We know how to formulate equations of different degree, and it is used a lot in real life, but the question arises, is it always possible to convert a situation into an equation? Sometimes we get statements like, the number of Covid cases per day in Delhi has reached more than 10,000. This phrase “L</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">8 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/solving-linear-inequalities-word-problems/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Solving Linear Inequalities Word Problems </div> <div class="gfg-similar-read-item-subheading">We are well versed with equations in multiple variables. Linear Equations represent a point in a single dimension, a line in a two-dimensional, and a plane in a three-dimensional world. Solutions to linear inequalities represent a region of the Cartesian plane. It becomes essential for us to know ho</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">11 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li></ul></div><div class="leftbar-dropdown"><h2 class="dropdown-title">Chapter 7: Permutations and Combinations<span class="gfg-icon gfg-icon_arrow-down-thin"></span></h2><ul style="display: none;"><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/fundamental-principle-of-counting/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Fundamental Principle of Counting </div> <div class="gfg-similar-read-item-subheading">Fundamental Principle of Counting is the basic principle that helps us to count large numbers in a non-tedious way. Suppose we have to guess the pin of a three-digit code so the number of ways we can guess is 1000 this can be seen as the pin can be, 000, 001, 002, ....., 999 any number between 000,</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">11 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/permutation/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Permutation </div> <div class="gfg-similar-read-item-subheading">In Mathematics, Permutation is defined as a mathematical concept that determines the number of possible arrangements for a specific set of elements. therefore, it plays a big role in computer science, cryptography, and operations research. For example, take a set {1, 2, 3}. All Permutations taking a</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">15+ min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/combinations/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Combinations </div> <div class="gfg-similar-read-item-subheading">Combination is a way of choosing items from a set, (unlike permutations) when the order of selection doesn't matter. In smaller cases, it's possible to count the number of combinations. Combination refers to the mixture of n things taken k at a time without repetition. Example: For set S = {a, b, c}</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">10 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li></ul></div><div class="leftbar-dropdown"><h2 class="dropdown-title">Chapter 8: Binomial Theorem<span class="gfg-icon gfg-icon_arrow-down-thin"></span></h2><ul style="display: none;"><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/binomial-theorem/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Binomial Theorem </div> <div class="gfg-similar-read-item-subheading">Binomial Theorem is a theorem that is used to find the expansion of algebraic identity (ax + by)n. We can easily find the expansion of (x + y)2, (x + y)3, and others but finding the expansion of (x + y)21 is a tedious task and this task can easily be achieved using the Binomial Theorem or Binomial E</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">15+ min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/pascals-triangle/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Pascal's Triangle </div> <div class="gfg-similar-read-item-subheading">Pascal's Triangle is a numerical pattern arranged in a triangular form. This triangle provides the coefficients for the expansion of any binomial expression, with numbers organized in a way that they form a triangular shape. i.e. the second row in Pascal's triangle represents the coefficients in (x+</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">12 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li></ul></div><div class="leftbar-dropdown"><h2 class="dropdown-title">Chapter 9: Sequences and Series<span class="gfg-icon gfg-icon_arrow-down-thin"></span></h2><ul style="display: none;"><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/sequences-and-series/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Sequences and Series </div> <div class="gfg-similar-read-item-subheading">A sequence is an ordered list of numbers following a specific rule. Each number in a sequence is called a "term." The order in which terms are arranged is crucial, as each term has a specific position, often denoted as an, where n indicates the position in the sequence. For example: 2, 5, 8, 11, 14</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">7 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/general-and-middle-terms-binomial-theorem-class-11-maths/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">General and Middle Terms - Binomial Theorem - Class 11 Maths </div> <div class="gfg-similar-read-item-subheading">Binomial theorem or expansion describes the algebraic expansion of powers of a binomial. According to this theorem, it is possible to expand the polynomial "(a + b)n" into a sum involving terms of the form "axzyc", the exponents z and c are non-negative integers where z + c = n, and the coefficient</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">7 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/arithmetic-series/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Arithmetic Series </div> <div class="gfg-similar-read-item-subheading">An arithmetic series is the sum of the terms of an arithmetic sequence, where an arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. Or we can say that an arithmetic progression can be defined as a sequence of numbers in which for every pair of</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">5 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/arithmetic-sequence/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Arithmetic Sequence </div> <div class="gfg-similar-read-item-subheading">In the arithmetic sequence, the absolute difference between one term and the next term is constant. In other words, a sequence a1, a2, . . ., an is called an arithmetic sequence or arithmetic progression if an+1 - an = d where d is constant and it is the common difference of the sequence. Let's make</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">8 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/what-is-geometric-progression/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Geometric Progression (GP) | Formula, Sums and Properties </div> <div class="gfg-similar-read-item-subheading">Geometric Progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio. Natural Examples of GPPowers of a Number : For example powers of 2: 1, 2, 4, 8, 16, 21, .... form a GP with common ratio 2 All Same Num</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">12 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/geometric-series/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Geometric Series </div> <div class="gfg-similar-read-item-subheading">In a Geometric Series, every next term is the multiplication of its Previous term by a certain constant, and depending upon the value of the constant, the Series may increase or decrease. Geometric Sequence is given as: a, ar, ar2, ar3, ar4,..... {Infinite Sequence}a, ar, ar2, ar3, ar4, ....... arn</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">3 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/arithmetic-progression-and-geometric-progression/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Arithmetic Progression and Geometric Progression </div> <div class="gfg-similar-read-item-subheading">Arithmetic Progression and Geometric Progression: The word "sequence" in English means a collection of some numbers or objects in such a way that it has a first member, a second member, and so on. Sequences can be of anything, for example. - January, February, .... is the sequence of months in a yea</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">11 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/special-series-sequences-and-series-class-11-maths/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Special Series in Maths - Sequences and Series | Class 11 Maths </div> <div class="gfg-similar-read-item-subheading">Special Series: A series can be defined as the sum of all the numbers of the given sequence. The sequences are finite as well as infinite. In the same way, the series can also be finite or infinite. For example, consider a sequence as 1, 3, 5, 7, … Then the series of these terms will be 1 + 3 + 5 +</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">10 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li></ul></div><div class="leftbar-dropdown"><h2 class="dropdown-title">Chapter 10: Straight Lines<span class="gfg-icon gfg-icon_arrow-down-thin"></span></h2><ul style="display: none;"><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/slope-of-line/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Slope of a Line </div> <div class="gfg-similar-read-item-subheading">Slope of a Line is the measure of the steepness of a line a surface or a curve whichever is the point of consideration. The slope of a Line is a fundamental concept in the stream of calculus or coordinate geometry or we can say the slope of a line is fundamental to the complete mathematics subject.</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">12 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/introduction-to-two-variable-linear-equations-in-straight-lines/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Introduction to Two-Variable Linear Equations in Straight Lines </div> <div class="gfg-similar-read-item-subheading">Lines are the most basic configuration in geometry. Many other geometrical shapes can be obtained from lines. Lines are referred 1-Dimensional. We can obtain higher dimensional shapes using lines. Let's understand the lines in depth. Let's say, we have two sets as follows, x = {1, 2, 3, 4, 5, 6, 7}</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">6 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/forms-of-two-variable-linear-equations-straight-lines-class-11-maths/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Forms of Two-Variable Linear Equations - Straight Lines | Class 11 Maths </div> <div class="gfg-similar-read-item-subheading">Line is the simplest geometrical shape. It has no endpoints and extends in both directions till infinity. The word “straight” simply means without “bend”. The gradient between any two point on the line is same. Hence, we can say that, if the gradient between any two points on the line is same, then</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">5 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/point-slope-form-straight-lines-class-11-maths/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Point-slope Form - Straight Lines | Class 11 Maths </div> <div class="gfg-similar-read-item-subheading">There are several forms to represent the equation of a straight line on the two-dimensional coordinate plane. Three major of them are point-slope form, slope-intercept form, and general or standard form. The point-slope form includes the slope of the straight line and a point on the line as the name</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">9 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/slope-intercept-form/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Slope Intercept Form </div> <div class="gfg-similar-read-item-subheading">The slope-intercept formula is one of the formulas used to find the equation of a line. The slope-intercept formula of a line with slope m and y-intercept b is, y = mx + b. Here (x, y) is any point on the line. It represents a straight line that cuts both axes. Slope intercept form of the equation i</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">9 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/writing-slope-intercept-equations/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Writing Slope-Intercept Equations </div> <div class="gfg-similar-read-item-subheading">Straight-line equations, also known as "linear" equations, have simple variable expressions with no exponents and graph as straight lines. A straight-line equation is one that has only two variables: x and y, rather than variables like y2 or √x. Because it contains information about these two proper</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">10 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/standard-form-of-a-straight-line/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Standard Form of a Straight Line </div> <div class="gfg-similar-read-item-subheading">There are several forms available to represent the equation of a straight line on the 2-dimensional coordinate plane, out of several forms three major forms are point-slope form, slope-intercept form, and general or standard form. The general or standard form is a linear equation where the degree of</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">12 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/x-and-y-intercept/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">X and Y Intercept </div> <div class="gfg-similar-read-item-subheading">The x-intercept is the point at which the graph crosses the x-axis. At this point, the y-coordinate is zero. The y-intercept is the point at which the graph crosses the y-axis. At this point, the x-coordinate is zero. In this article, we will explore the definition of intercepts including both x and</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">9 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/graphing-slope-intercept-equations-straight-lines-class-11-maths/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Graphing Slope-Intercept Equations - Straight Lines | Class 11 Maths </div> <div class="gfg-similar-read-item-subheading">To graph a straight line we need at least two points that lie on the straight line. From the slope-intercept form of the given straight line, we can calculate two points on the line very easily using the information present in the equation. Consider a straight line with slope m and y-intercept c. We</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">7 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li></ul></div><div class="leftbar-dropdown"><h2 class="dropdown-title">Chapter 11: Conic Sections<span class="gfg-icon gfg-icon_arrow-down-thin"></span></h2><ul style="display: none;"><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/conic-sections/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Conic Sections </div> <div class="gfg-similar-read-item-subheading">Conic sections also called the section of the cone are fundamental curves in mathematics formed by the intersection of a plane and a double-napped cone. Thus, the name Conic Sections because they are the section of a cone. By the intersection of a cone with a plane, we formed four different types of</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">14 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/equation-of-a-circle/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Equation of a Circle </div> <div class="gfg-similar-read-item-subheading">Equation for a circle in standard form is written as (x - x1 )2 + (y - y1 )2 = r2. Here, (x1, y1) is the centre of the circle. In this article, we will learn about equation of circle, various forms of equation of circle, how to find equation of circle, and others in detail. The image below shows a c</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">15+ min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/focus-and-directrix-of-a-parabola/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Focus and Directrix of a Parabola </div> <div class="gfg-similar-read-item-subheading">In mathematics, a parabola is the locus of a point that moves in a plane where its distance from a fixed point known as the focus is always equal to the distance from a fixed straight line known as directrix in the same plane. Or in other words, a parabola is a plane curve that is almost in U shape</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">8 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/ellipse/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Ellipse </div> <div class="gfg-similar-read-item-subheading">An ellipse is a geometric shape that looks like a stretched circle or a flattened oval. It is defined by two points called the foci and a constant sum of distances from these foci to any point on the ellipse. This sum is constant for every point on the ellipse, which gives it its unique shape. Ellip</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">14 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/hyperbola/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Hyperbola </div> <div class="gfg-similar-read-item-subheading">Conic sections are used in everyday life, from a guitar, flyover to a football. Everything has a curve that belongs to the curves of conic sections. There are four types of conic sections - circles, parabola, ellipse, and hyperbola. Hyperbola and Ellipse are similar kinds of conic sections. A hyperb</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">7 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/identifying-conic-sections-from-their-equation/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Identifying Conic Sections from their Equation </div> <div class="gfg-similar-read-item-subheading">Conic section is defined as a locus of a point (P) in a plane which moves in such a way, that the ratio of its distances from a fixed point (called focus of conic, say S) and from a fixed line (called directrix of conic, say y-axis L) is constant (called the eccentricity of conic, denoted by 'e') i.</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">11 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li></ul></div><div class="leftbar-dropdown"><h2 class="dropdown-title">Chapter 12: Introduction to Three-dimensional Geometry<span class="gfg-icon gfg-icon_arrow-down-thin"></span></h2><ul style="display: none;"><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/coordinate-axes-and-coordinate-planes-in-3d-space/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Coordinate Axes and Coordinate Planes in 3D space </div> <div class="gfg-similar-read-item-subheading">In a plane, we know that we need two mutually perpendicular lines to locate the position of a point. These lines are called coordinate axes of the plane and the plane is usually called the Cartesian plane. But in real life, we do not have such a plane. In real life, we need some extra information su</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">6 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/3d-distance-formula/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">3D Distance Formula </div> <div class="gfg-similar-read-item-subheading">3D Distance Formula: Distance Formula in 3D calculates the distance between two points, a point and a line, and a point and a plane in three-dimensional coordinates as well as a two-dimensional Cartesian Plane. This article deals with the distance formula of points in three-dimensional space. In thr</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">10 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li></ul></div><div class="leftbar-dropdown"><h2 class="dropdown-title">Chapter 13: Limits and Derivatives<span class="gfg-icon gfg-icon_arrow-down-thin"></span></h2><ul style="display: none;"><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/limits/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Limits in Calculus </div> <div class="gfg-similar-read-item-subheading">Limits in maths are defined as the values approaching the output for the given input values of a function. Limits are used in calculus and mathematical analysis for finding the derivatives of the function. They are also used to define the continuity of the function. In this article, we will learn th</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">15 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/formal-definition-of-limits/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Formal Definition of Limits </div> <div class="gfg-similar-read-item-subheading">Intuitively limit for a function f(x) at a point x, means the value that the function seems to be approaching as one moves towards the point x. The understanding of these concepts leads to the development of some fundamental concepts of calculus - continuity, derivatives, and integrals. There are th</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">10 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/strategy-in-finding-limits/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Strategy in Finding Limits </div> <div class="gfg-similar-read-item-subheading">Limits have been really useful in the field of calculus, they become a solid foundation for defining many concepts like continuity, differentiability, integrals, and derivatives. These concepts further help us analyze a lot of functions and their behavior in calculus. Limits have been the foundation</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">10 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/determining-limits-using-algebraic-manipulation/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Determining Limits using Algebraic Manipulation </div> <div class="gfg-similar-read-item-subheading">Limits give us the power to approximate functions and see the values they are approaching. Limit is not the value of the function at a particular point. It is the value which the function is approaching as one moves towards the given point. There are many ways to solve the limits, often limits are e</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">7 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/limits-of-trigonometric-functions/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Limits of Trigonometric Functions </div> <div class="gfg-similar-read-item-subheading">Trigonometry is one of the most important branch of Mathematics. We know that there are six trigonometric functions and the limit of trigonometric is the limit taken to each trigonometric function. We can easily find the limit of trigonometric functions and the limit of the trigonometric function ma</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">10 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/properties-of-limits/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Properties of Limits </div> <div class="gfg-similar-read-item-subheading">Limits form the foundation of the theory of calculus. Limits of the functions are used to define the derivatives of the functions, check the continuity in functions, and so on. Intuitively value of the limit of a function at a particular point gives us an idea about the approaching value of the func</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">7 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/limits-by-direct-substitution/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Limits by Direct Substitution </div> <div class="gfg-similar-read-item-subheading">Limits are building blocks of calculus. They are the values that a function seems to be taking on while we reach a particular point. They help is calculating the rate of change of the functions. The concept of derivatives has been defined with limits. They also help us define the concepts of continu</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">7 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/estimating-limits-from-graphs/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Estimating Limits from Graphs </div> <div class="gfg-similar-read-item-subheading">The concept of limits has been around for thousands of years. Earlier mathematicians in ancient civilizations used limits to approximate the area of a circle. However the formal concept was not around till the 19th century. This concept is essential to calculus and serves as a building block for ana</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">10 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/estimating-limits-from-tables/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Estimating Limits from Tables </div> <div class="gfg-similar-read-item-subheading">Limits tell us a lot about function behavior. They help mathematicians and engineers reason about the function their behavior and their properties. They form the basis for almost every important concept in calculus. Limits help us estimate the values function seems to be taking at a particular point</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">9 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/sandwich-theorem/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Sandwich Theorem </div> <div class="gfg-similar-read-item-subheading">Sandwich Theorem also called Sandwich Rule or Squeeze Theorem, is an important theorem in calculus involving limits and it is used to find the limit of some functions when the normal methods of finding the limit fail. Suppose we have to find the limit such that limx->a f(x) gives an indeterminant</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">8 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/derivatives/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Derivatives | First and Second Order Derivatives, Formulas and Examples </div> <div class="gfg-similar-read-item-subheading">Derivatives: In mathematics, a Derivative represents the rate at which a function changes as its input changes. It measures how a function's output value moves as its input value nudges a little bit. This concept is a fundamental piece of calculus. It is used extensively across science, engineering,</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">15+ min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/average-and-instantaneous-rate-of-change/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Average and Instantaneous Rate of Change </div> <div class="gfg-similar-read-item-subheading">The average rate of change represents the total change in one variable in relation to the total change of another variable. Instantaneous rate of change, or derivative, measures the specific rate of change of one variable in relation to a specific, infinitesimally small change in the other variable.</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">9 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/algebra-of-derivative-of-functions/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Algebra of Derivative of Functions </div> <div class="gfg-similar-read-item-subheading">Derivatives are an integral part of calculus. They measure the rate of change in any quantity. Suppose there is a water tank from which water is leaking. A local engineer is asked to measure the time in which the water tank will become empty. In such a scenario, the engineer needs to know two things</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">8 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/product-rule-derivatives/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Product Rule in Derivatives </div> <div class="gfg-similar-read-item-subheading">Product Rule is the rule that is used to find the derivative of the function that is expressed as the product of two functions. The product rule in calculus is the fundamental rule and is used to find the derivative of the functions. Product Rule of the calculus is proved using the concept of limit</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">9 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/quotient-rule/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Quotient Rule: Formula, Proof, Definition, Examples </div> <div class="gfg-similar-read-item-subheading">Quotient Rule is a method for finding the derivative of a function that is the quotient of two other functions. It is a method used for differentiating problems where one function is divided by another. We use the quotient rule when we have to find the derivative of a function of the form: f(x)/g(x)</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">7 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/derivatives-of-polynomial-functions/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Derivatives of Polynomial Functions </div> <div class="gfg-similar-read-item-subheading">Derivatives are used in Calculus to measure the rate of change of a function with respect to a variable. The use of derivatives is very important in Mathematics. It is used to solve many problems in mathematics like to find out maxima or minima of a function, slope of a function, to tell whether a f</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">3 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/power-rule/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Power Rule </div> <div class="gfg-similar-read-item-subheading">Power Rule is a fundamental rule in the calculation of derivatives that helps us find the derivatives of functions with exponents. Exponents can take any form, including any function itself. With the help of the Power Rule, we can differentiate polynomial functions, functions with variable exponents</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">7 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/application-of-derivatives/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Application of Derivatives </div> <div class="gfg-similar-read-item-subheading">Derivatives are a fundamental concept in calculus. They measure how a function changes as its input changes. This makes Derivatives very useful in various fields. For example, derivatives help in understanding motion, growth, and change in physical, economic, and engineering systems. They are used t</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">15+ min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/applications-of-power-rule/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Applications of Power Rule </div> <div class="gfg-similar-read-item-subheading">Differentiation of a function is breaking up the function into a smaller and much simpler version of itself. Derivative of any function is differentiating it with respect to the variable present in it, in easier words, derivatives are the rate of change of any function with respect to the variable p</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">7 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li></ul></div><div class="leftbar-dropdown"><h2 class="dropdown-title">Chapter 14: Mathematical Reasoning<span class="gfg-icon gfg-icon_arrow-down-thin"></span></h2><ul style="display: none;"><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/statements-mathematical-reasoning/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Statements - Mathematical Reasoning </div> <div class="gfg-similar-read-item-subheading">Statements - Mathematical Reasoning: The study of logic through mathematical symbols is called mathematical reasoning. Mathematical logic is also known as Boolean logic. In other words, in mathematical reasoning, we determine the statement's truth value. Table of Content What is Mathematical Reasoni</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">10 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/conditional-statements-implications-mathematical-reasoning-class-11-maths/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Conditional Statements & Implications - Mathematical Reasoning | Class 11 Maths </div> <div class="gfg-similar-read-item-subheading">Generally, Conditional statements are the if-then statement in which p is called a hypothesis(or antecedent or premise) and q is called a conclusion( or consequence). Conditional Statements symbolized by p, q. A Conditional statement p -> q is false when p is true and q is false, and true otherwi</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">5 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li></ul></div><div class="leftbar-dropdown"><h2 class="dropdown-title">Chapter 15: Statistics<span class="gfg-icon gfg-icon_arrow-down-thin"></span></h2><ul style="display: none;"><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/measures-of-spread-range-variance-and-standard-deviation/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Measures of Spread - Range, Variance, and Standard Deviation </div> <div class="gfg-similar-read-item-subheading">Collecting the data and representing it in form of tables, graphs, and other distributions is essential for us. But, it is also essential that we get a fair idea about how the data is distributed, how scattered it is, and what is the mean of the data. The measures of the mean are not enough to descr</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">9 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/mean-absolute-deviation/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Mean Absolute Deviation </div> <div class="gfg-similar-read-item-subheading">Mean Absolute Deviation is one of the metrics of statistics that helps us find out the average spread of the data i.e., Mean Absolute Deviation shows the average distance of the observation of the dataset from the mean of the dataset. It is helpful in the analysis of data and understanding of the da</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">8 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/measures-of-central-tendency/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Measures of Central Tendency in Statistics </div> <div class="gfg-similar-read-item-subheading">Central Tendencies in Statistics are the numerical values that are used to represent mid-value or central value a large collection of numerical data. These obtained numerical values are called central or average values in Statistics. A central or average value of any statistical data or series is th</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">10 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/difference-between-mean-median-and-mode-with-examples/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Difference Between Mean, Median, and Mode with Examples </div> <div class="gfg-similar-read-item-subheading">The measures of central tendency most widely used in the field of descriptive statistics are mean, median, and mode. All the measures provide a different view of the center of the data and ensures that the information is well summarized and interpreted. The mean, or commonly known as the average val</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">5 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/frequency-distribution/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Frequency Distribution - Table, Graphs, Formula </div> <div class="gfg-similar-read-item-subheading">Frequency Distribution is a tool in statistics that helps us organize the data and also helps us reach meaningful conclusions. It tells us how often any specific values occur in the dataset. A frequency distribution in a tabular form organizes data by showing the frequencies (the number of times val</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">12 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/variance-and-standard-deviation/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Variance and Standard Deviation </div> <div class="gfg-similar-read-item-subheading">Variance and Standard Deviation are the important measures used in Mathematics and Statics to find the meaning from a large set of data. The different formulas for Variance and Standard Deviation are highly used in mathematics to determine the trends of various values in mathematics. Variance is the</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">8 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li></ul></div><div class="leftbar-dropdown"><h2 class="dropdown-title">Chapter 16: Probability<span class="gfg-icon gfg-icon_arrow-down-thin"></span></h2><ul style="display: none;"><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/random-experiment-probability/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Random Experiment - Probability </div> <div class="gfg-similar-read-item-subheading">In a cricket match, before the game begins. Two captains go for a toss. Tossing is an activity of flipping a coin and checking the result as either “Head” or “Tail”. Similarly, tossing a die gives us a number from 1 to 6. All these activities are examples of experiments. An activity that gives us a</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">12 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/types-of-events-in-probability/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Types of Events in Probability </div> <div class="gfg-similar-read-item-subheading">Whenever an experiment is performed whose outcomes cannot be predicted with certainty, it is called a random experiment. In such cases, we can only measure which of the events is more likely or less likely to happen. This likelihood of events is measured in terms of probability and events refer to t</div> </div> <div class="reading-time"> <i class="reading-time-icon" data-gfg-action="readingtime"></i> <span style="color: var(--recommendation-card-text-color);font-size: 14px;font-weight: 500;line-height: 17px;">13 min read</span> </div> </a><hr style="background:var(--gfg-selected-color); width:70%; margin: 0 auto;"></hr></li><li class="similarReadDropdownItem"><a href="https://www.geeksforgeeks.org/events-in-probability/?ref=lbp"><div class="gfg-similar-read-item-content"> <div class="gfg-similar-read-item-heading">Events in Probability </div> <div class="gfg-similar-read-item-subheading">Events in Probability- In Probability, an event can be defined as any outcome or set of outcomes from a random experiment. In other words, an event in probability is the subset of the respective sample space. Example:1. 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align-items: center; border-radius: 0px 10px 10px 0px; height: 80px; width: calc(100% - 80px); } .popup-main .exp-card1 { justify-content: flex-start; background: var(--write-experience-card-1); } .popup-main .exp-card2 { justify-content: flex-start; background: var(--write-experience-card-2); } .popup-main .exp-card3 { justify-content: flex-start; background: var(--write-experience-card-3); } .popup-main .exp-card4 { justify-content: flex-start; background: var(--write-experience-card-4); } .popup-main .exp-card5 { justify-content: flex-start; background: var(--write-experience-card-5); } .popup-main .exp-card6 { justify-content: flex-start; background: var(--write-experience-card-6); } .popup-main span { font-family: var(--font-primary); font-size: 14px; font-style: normal; font-weight: 500; line-height: normal; padding-left: 5px; padding-right: 5px; } #popup { display: none; background-color: var(--write-modal-background); padding: 20px; text-align: center; position: fixed; top: 50%; left: 50%; transform: translate(-50%, -50%); z-index: 1026; width: min(650px, calc(100% - 20px)); } .link-container-write{ display:flex; flex-direction:column; } .link-container-write > a { margin:7px 0px; } .phrase__container{ color:var(--color-gfg); border-radius: 6px; width: 100%; background-color:var(--write-redirect-container); padding: 15px; text-align: justify; font-size:14px; } .phrase__container:hover { color:var(--color-gfg) !important; background-color:var(--write-redirect-container-hover); } </style> <div class="popup-container"> <div id="popup" accesskey="" class="popup-main"> <div class="popup-heading"> <h2>What kind of Experience do you want to share?</h2> <div class="close-icon share-experience-modal-close"></div> </div>  <div class="experience-card"> <a class="exp-card5" href= "https://write.geeksforgeeks.org/posts-new?cid=e8fc46fe-75e7-4a4b-be3c-0c862d655ed0" target="_blank"> <div class="icon5"> <div class="icon5-image"></div> </div> <div class="exp-card5-text"> <span style="color: #000;">Interview Experiences</span> </div> </a> <a class="exp-card1" href="https://write.geeksforgeeks.org/posts-new?cid=82536bdb-84e6-4661-87c3-e77c3ac04ede" target="_blank"> <div class="icon1"> <div class="icon1-image"></div> </div> <div class="exp-card1-text"> <span style="color: #000;">Admission Experiences</span> </div> </a> <a class="exp-card6" href= "https://write.geeksforgeeks.org/posts-new?cid=5219b0b2-7671-40a0-9bda-503e28a61c31" target="_blank"> <div class="icon6"> <div class="icon6-image"></div> </div> <div class="exp-card6-text"> <span style="color: #000;">Career Journeys</span> </div> </a> <a class="exp-card2" href="https://write.geeksforgeeks.org/posts-new?cid=22ae3354-15b6-4dd4-a5b4-5c7a105b8a8f" target="_blank"> <div class="icon2"> <div class="icon2-image"></div> </div> <div class="exp-card2-text"> <span style="color: #000;">Work Experiences</span> </div> </a> <a class="exp-card3" href= "https://write.geeksforgeeks.org/posts-new?cid=c5e1ac90-9490-440a-a5fa-6180c87ab8ae" target="_blank"> <div class="icon3"> <div class="icon3-image"></div> </div> <div class="exp-card3-text"> <span style="color: #000;">Campus Experiences</span> </div> </a> <a class="exp-card4" href= "https://write.geeksforgeeks.org/posts-new?cid=5ebb8fe9-b980-4891-af07-f2d62a9735f2" target="_blank"> <div class="icon4"> <div class="icon4-image"></div> </div> <div class="exp-card4-text"> <span style="color: #000;">Competitive Exam Experiences</span> </div> </a> </div>  </div> <div id="overlay" onclick="toggleExperiencePopup()" style="display: block;min-height: 100vh;min-width: 100vw;position: fixed;top: 0;bottom: 0;right: 0;left: 0;display: none;justify-content: center;align-items: center;background: rgba(0,0,0,.702);z-index: 1025 !important;backdrop-filter: blur(4px); -webkit-backdrop-filter: blur(4px);word-break: keep-all;"></div> </div> <script> $('.share-experience-modal').click(function(e){ e.preventDefault(); var link = $(this).attr('href'); toggleExperiencePopup(link); }); function toggleExperiencePopup(link) { var popup = document.getElementById("popup"); var overlay = document.getElementById("overlay"); if (window.innerWidth < 992) { window.location.href = link; } else { popup.style.display = (popup.style.display === "block") ? "none" : "block"; overlay.style.display = (overlay.style.display === "block") ? "none" : "block"; } } $('.share-experience-modal-close').click(function(e){ var popup = document.getElementById("popup"); popup.style.display = 'none'; var overlay = document.getElementById("overlay"); overlay.style.display = 'none'; }); </script> </div><script type="text/javascript"> $(window).on('load', function() { (function(c,l,a,r,i,t,y){ c[a]=c[a]||function(){(c[a].q=c[a].q||[]).push(arguments)}; t=l.createElement(r);t.async=1;t.src="https://www.clarity.ms/tag/"+i; y=l.getElementsByTagName(r)[0];y.parentNode.insertBefore(t,y); })(window, document, "clarity", "script", "ayc7ypwwuk"); }); </script>  </body> </html>

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