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<a class="vector-toc-link" href="#Concepts_in_discrete_mathematics"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Concepts in discrete mathematics</span> </div> </a> <button aria-controls="toc-Concepts_in_discrete_mathematics-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Concepts in discrete mathematics subsection</span> </button> <ul id="toc-Concepts_in_discrete_mathematics-sublist" class="vector-toc-list"> <li id="toc-Sets" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sets"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Sets</span> </div> </a> <ul id="toc-Sets-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Functions"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Functions</span> </div> </a> <ul id="toc-Functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Arithmetic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Arithmetic"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Arithmetic</span> </div> </a> <ul id="toc-Arithmetic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Elementary_algebra" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Elementary_algebra"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Elementary algebra</span> </div> </a> <ul id="toc-Elementary_algebra-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mathematical_relations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mathematical_relations"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Mathematical relations</span> </div> </a> <ul id="toc-Mathematical_relations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mathematical_phraseology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mathematical_phraseology"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.6</span> <span>Mathematical phraseology</span> </div> </a> <ul id="toc-Mathematical_phraseology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Combinatorics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Combinatorics"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.7</span> <span>Combinatorics</span> </div> </a> <ul id="toc-Combinatorics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Probability" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Probability"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.8</span> <span>Probability</span> </div> </a> <ul id="toc-Probability-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Propositional_logic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Propositional_logic"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.9</span> <span>Propositional logic</span> </div> </a> <ul id="toc-Propositional_logic-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Mathematicians_associated_with_discrete_mathematics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Mathematicians_associated_with_discrete_mathematics"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Mathematicians associated with discrete mathematics</span> </div> </a> <ul id="toc-Mathematicians_associated_with_discrete_mathematics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main 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<div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Overview of and topical guide to discrete mathematics</div> <p><b><a href="/wiki/Discrete_mathematics" title="Discrete mathematics">Discrete mathematics</a></b> is the study of <a href="/wiki/Mathematics" title="Mathematics">mathematical</a> <a href="/wiki/Mathematical_structure" title="Mathematical structure">structures</a> that are fundamentally <a href="/wiki/Discrete_space" title="Discrete space">discrete</a> rather than <a href="/wiki/Continuous_function" title="Continuous function">continuous</a>. In contrast to <a href="/wiki/Real_number" title="Real number">real numbers</a> that have the property of varying "smoothly", the objects studied in discrete mathematics – such as <a href="/wiki/Integer" title="Integer">integers</a>, <a href="/wiki/Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graphs</a>, and statements in <a href="/wiki/Mathematical_logic" title="Mathematical logic">logic</a><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> – do not vary smoothly in this way, but have distinct, separated values.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Discrete mathematics, therefore, excludes topics in "continuous mathematics" such as <a href="/wiki/Calculus" title="Calculus">calculus</a> and <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">analysis</a>. </p><p>Included below are many of the standard terms used routinely in university-level courses and in research papers. This is not, however, intended as a complete list of mathematical terms; just a selection of typical <i><a href="/wiki/Term_of_art" class="mw-redirect" title="Term of art">terms of art</a></i> that may be encountered. </p> <ul><li><a href="/wiki/Logic" title="Logic">Logic</a> – Study of correct reasoning</li> <li><a href="/wiki/Modal_logic" title="Modal logic">Modal logic</a> – Type of formal logic</li> <li><a href="/wiki/Set_theory" title="Set theory">Set theory</a> – Branch of mathematics that studies sets</li> <li><a href="/wiki/Number_theory" title="Number theory">Number theory</a> – Mathematics of integer properties</li> <li><a href="/wiki/Combinatorics" title="Combinatorics">Combinatorics</a> – Branch of discrete mathematics</li> <li><a href="/wiki/Finite_mathematics" title="Finite mathematics">Finite mathematics</a> – Syllabus in college and university mathematics</li> <li><a href="/wiki/Graph_theory" title="Graph theory">Graph theory</a> – Area of discrete mathematics</li> <li><a href="/wiki/Digital_geometry" title="Digital geometry">Digital geometry</a> – Deals with digitized models or images of objects of the 2D or 3D Euclidean space</li> <li><a href="/wiki/Digital_topology" title="Digital topology">Digital topology</a> – Properties of 2D or 3D digital images that correspond to classic topological properties</li> <li><a href="/wiki/Algorithmics" class="mw-redirect" title="Algorithmics">Algorithmics</a> – Sequence of operations for a task<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a> – Scientific study of digital information</li> <li><a href="/wiki/Computability" title="Computability">Computability</a> – Ability to solve a problem in an effective manner</li> <li><a href="/wiki/Computational_complexity_theory" title="Computational complexity theory">Computational complexity theory</a> – Inherent difficulty of computational problems</li> <li><a href="/wiki/Probability_theory" title="Probability theory">Probability theory</a> – Branch of mathematics concerning probability</li> <li><a href="/wiki/Probability" title="Probability">Probability</a> – Branch of mathematics concerning chance and uncertainty</li> <li><a href="/wiki/Markov_chain" title="Markov chain">Markov chains</a> – Random process independent of past history</li> <li><a href="/wiki/Linear_algebra" title="Linear algebra">Linear algebra</a> – Branch of mathematics</li> <li><a href="/wiki/Function_(mathematics)" title="Function (mathematics)">Functions</a> – Association of one output to each input</li> <li><a href="/wiki/Partially_ordered_set" title="Partially ordered set">Partially ordered set</a> – Mathematical set with an ordering</li> <li><a href="/wiki/Mathematical_proof" title="Mathematical proof">Proofs</a> – Reasoning for mathematical statements</li> <li><a href="/wiki/Relation_(mathematics)" title="Relation (mathematics)">Relation</a> – Relationship between two sets, defined by a set of ordered pairs</li></ul> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Discrete_mathematical_disciplines">Discrete mathematical disciplines</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Outline_of_discrete_mathematics&action=edit&section=1" title="Edit section: Discrete mathematical disciplines"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For further reading in discrete mathematics, beyond a basic level, see these pages. Many of these disciplines are closely related to <a href="/wiki/Computer_science" title="Computer science">computer science</a>. </p> <ul><li><a href="/wiki/Automata_theory" title="Automata theory">Automata theory</a> – Study of abstract machines and automata</li> <li><a href="/wiki/Coding_theory" title="Coding theory">Coding theory</a> – Study of the properties of codes and their fitness</li> <li><a href="/wiki/Combinatorics" title="Combinatorics">Combinatorics</a> – Branch of discrete mathematics</li> <li><a href="/wiki/Computational_geometry" title="Computational geometry">Computational geometry</a> – Branch of computer science</li> <li><a href="/wiki/Digital_geometry" title="Digital geometry">Digital geometry</a> – Deals with digitized models or images of objects of the 2D or 3D Euclidean space</li> <li><a href="/wiki/Discrete_geometry" title="Discrete geometry">Discrete geometry</a> – Branch of geometry that studies combinatorial properties and constructive methods</li> <li><a href="/wiki/Graph_theory" title="Graph theory">Graph theory</a> – Area of discrete mathematics a study of <a href="/wiki/Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graphs</a> – Vertices connected in pairs by edges</li> <li><a href="/wiki/Mathematical_logic" title="Mathematical logic">Mathematical logic</a> – Subfield of mathematics</li> <li><a href="/wiki/Discrete_optimization" title="Discrete optimization">Discrete optimization</a> – Branch of mathematical optimization</li> <li><a href="/wiki/Set_theory" title="Set theory">Set theory</a> – Branch of mathematics that studies sets</li> <li><a href="/wiki/Number_theory" title="Number theory">Number theory</a> – Mathematics of integer properties</li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a> – Scientific study of digital information</li> <li><a href="/wiki/Game_theory" title="Game theory">Game theory</a> – Mathematical models of strategic interactions</li></ul> <div class="mw-heading mw-heading2"><h2 id="Concepts_in_discrete_mathematics">Concepts in discrete mathematics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Outline_of_discrete_mathematics&action=edit&section=2" title="Edit section: Concepts in discrete mathematics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Sets">Sets</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Outline_of_discrete_mathematics&action=edit&section=3" title="Edit section: Sets"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Set_(mathematics)" title="Set (mathematics)">Set (mathematics)</a> – Collection of mathematical objects <ul><li><a href="/wiki/Element_(mathematics)" title="Element (mathematics)">Element (mathematics)</a> – Any one of the distinct objects that make up a set in set theory</li> <li><a href="/wiki/Venn_diagram" title="Venn diagram">Venn diagram</a> – Diagram that shows all possible logical relations between a collection of sets</li> <li><a href="/wiki/Empty_set" title="Empty set">Empty set</a> – Mathematical set containing no elements</li> <li><a href="/wiki/Subset" title="Subset">Subset</a> – Set whose elements all belong to another set</li> <li><a href="/wiki/Union_(set_theory)" title="Union (set theory)">Union (set theory)</a> – Set of elements in any of some sets <ul><li><a href="/wiki/Disjoint_union" title="Disjoint union">Disjoint union</a> – In mathematics, operation on sets</li></ul></li> <li><a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">Intersection (set theory)</a> – Set of elements common to all of some sets <ul><li><a href="/wiki/Disjoint_sets" title="Disjoint sets">Disjoint sets</a> – Sets with no element in common</li></ul></li> <li><a href="/wiki/Complement_(set_theory)" title="Complement (set theory)">Complement (set theory)</a> – Set of the elements not in a given subset</li> <li><a href="/wiki/Symmetric_difference" title="Symmetric difference">Symmetric difference</a> – Elements in exactly one of two sets</li></ul></li> <li><a href="/wiki/Ordered_pair" title="Ordered pair">Ordered pair</a> – Pair of mathematical objects</li> <li><a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a> – Mathematical set formed from two given sets</li> <li><a href="/wiki/Power_set" title="Power set">Power set</a> – Mathematical set of all subsets of a set</li> <li><a href="/wiki/Simple_theorems_in_the_algebra_of_sets" title="Simple theorems in the algebra of sets">Simple theorems in the algebra of sets</a></li> <li><a href="/wiki/Naive_set_theory" title="Naive set theory">Naive set theory</a> – Informal set theories</li> <li><a href="/wiki/Multiset" title="Multiset">Multiset</a> – Mathematical set with repetitions allowed</li></ul> <div class="mw-heading mw-heading3"><h3 id="Functions">Functions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Outline_of_discrete_mathematics&action=edit&section=4" title="Edit section: Functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Function_(mathematics)" title="Function (mathematics)">Function</a> – Association of one output to each input</li> <li><a href="/wiki/Domain_of_a_function" title="Domain of a function">Domain of a function</a> – Mathematical concept</li> <li><a href="/wiki/Codomain" title="Codomain">Codomain</a> – Target set of a mathematical function</li> <li><a href="/wiki/Range_of_a_function" title="Range of a function">Range of a function</a> – Subset of a function's codomain</li> <li><a href="/wiki/Image_(mathematics)" title="Image (mathematics)">Image (mathematics)</a> – Set of the values of a function</li> <li><a href="/wiki/Injective_function" title="Injective function">Injective function</a> – Function that preserves distinctness</li> <li><a href="/wiki/Surjection" class="mw-redirect" title="Surjection">Surjection</a> – Mathematical function such that every output has at least one input<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li> <li><a href="/wiki/Bijection" title="Bijection">Bijection</a> – One-to-one correspondence</li> <li><a href="/wiki/Function_composition" title="Function composition">Function composition</a> – Operation on mathematical functions</li> <li><a href="/wiki/Partial_function" title="Partial function">Partial function</a> – Function whose actual domain of definition may be smaller than its apparent domain</li> <li><a href="/wiki/Multivalued_function" title="Multivalued function">Multivalued function</a> – Generalized mathematical function</li> <li><a href="/wiki/Binary_function" title="Binary function">Binary function</a> – Function that takes two inputs</li> <li><a href="/wiki/Floor_function" class="mw-redirect" title="Floor function">Floor function</a> – Nearest integers from a number<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li> <li><a href="/wiki/Sign_function" title="Sign function">Sign function</a> – Mathematical function returning -1, 0 or 1</li> <li><a href="/wiki/Inclusion_map" title="Inclusion map">Inclusion map</a> – Set-theoretic function</li> <li><a href="/wiki/Pigeonhole_principle" title="Pigeonhole principle">Pigeonhole principle</a> – If there are more items than boxes holding them, one box must contain at least two items</li> <li><a href="/wiki/Relation_composition" class="mw-redirect" title="Relation composition">Relation composition</a> – Mathematical operation<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li> <li><a href="/wiki/Permutations" class="mw-redirect" title="Permutations">Permutations</a> – Mathematical version of an order change<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li> <li><a href="/wiki/Symmetry" title="Symmetry">Symmetry</a> – Mathematical invariance under transformations</li></ul> <div class="mw-heading mw-heading3"><h3 id="Arithmetic">Arithmetic</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Outline_of_discrete_mathematics&action=edit&section=5" title="Edit section: Arithmetic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Decimal" title="Decimal">Decimal</a> – Number in base-10 numeral system</li> <li><a href="/wiki/Binary_numeral_system" class="mw-redirect" title="Binary numeral system">Binary numeral system</a> – Number expressed in the base-2 numeral system<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li> <li><a href="/wiki/Divisor" title="Divisor">Divisor</a> – Integer that is a factor of another integer</li> <li><a href="/wiki/Division_by_zero" title="Division by zero">Division by zero</a> – Class of mathematical expression</li> <li><a href="/wiki/Indeterminate_form" title="Indeterminate form">Indeterminate form</a> – Expression in mathematical analysis</li> <li><a href="/wiki/Empty_product" title="Empty product">Empty product</a> – Result from multiplying no factors</li> <li><a href="/wiki/Euclidean_algorithm" title="Euclidean algorithm">Euclidean algorithm</a> – Algorithm for computing greatest common divisors</li> <li><a href="/wiki/Fundamental_theorem_of_arithmetic" title="Fundamental theorem of arithmetic">Fundamental theorem of arithmetic</a> – Integers have unique prime factorizations</li> <li><a href="/wiki/Modular_arithmetic" title="Modular arithmetic">Modular arithmetic</a> – Computation modulo a fixed integer</li> <li><a href="/wiki/Successor_function" title="Successor function">Successor function</a> – Elementary operation on a natural number</li></ul> <div class="mw-heading mw-heading3"><h3 id="Elementary_algebra">Elementary algebra</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Outline_of_discrete_mathematics&action=edit&section=6" title="Edit section: Elementary algebra"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Elementary_algebra" title="Elementary algebra">Elementary algebra</a> – Basic concepts of algebra </p> <ul><li><a href="/wiki/Left-hand_side_and_right-hand_side_of_an_equation" class="mw-redirect" title="Left-hand side and right-hand side of an equation">Left-hand side and right-hand side of an equation</a> – Mathematical nomenclature<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li> <li><a href="/wiki/Linear_equation" title="Linear equation">Linear equation</a> – Equation that does not involve powers or products of variables</li> <li><a href="/wiki/Quadratic_equation" title="Quadratic equation">Quadratic equation</a> – Polynomial equation of degree two</li> <li><a href="/wiki/Solution_point" class="mw-redirect" title="Solution point">Solution point</a> – Mathematical formula expressing equality<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li> <li><a href="/wiki/Arithmetic_progression" title="Arithmetic progression">Arithmetic progression</a> – Sequence of equally spaced numbers</li> <li><a href="/wiki/Recurrence_relation" title="Recurrence relation">Recurrence relation</a> – Pattern defining an infinite sequence of numbers</li> <li><a href="/wiki/Finite_difference" title="Finite difference">Finite difference</a> – Discrete analog of a derivative</li> <li><a href="/wiki/Difference_operator" class="mw-redirect" title="Difference operator">Difference operator</a> – Pattern defining an infinite sequence of numbers<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li> <li><a href="/wiki/Group_(mathematics)" title="Group (mathematics)">Groups</a> – Set with associative invertible operation</li> <li><a href="/wiki/Group_isomorphism" title="Group isomorphism">Group isomorphism</a> – Bijective group homomorphism</li> <li><a href="/wiki/Subgroups" class="mw-redirect" title="Subgroups">Subgroups</a> – Subset of a group that forms a group itself<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li> <li><a href="/wiki/Fermat%27s_little_theorem" title="Fermat's little theorem">Fermat's little theorem</a> – A prime p divides a^p–a for any integer a</li> <li><a href="/wiki/Cryptography" title="Cryptography">Cryptography</a> – Practice and study of secure communication techniques</li> <li><a href="/wiki/Faulhaber%27s_formula" title="Faulhaber's formula">Faulhaber's formula</a> – Expression for sums of powers</li></ul> <div class="mw-heading mw-heading3"><h3 id="Mathematical_relations">Mathematical relations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Outline_of_discrete_mathematics&action=edit&section=7" title="Edit section: Mathematical relations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Binary_relation" title="Binary relation">Binary relation</a> – Relationship between elements of two sets</li> <li><a href="/wiki/Heterogeneous_relation" class="mw-redirect" title="Heterogeneous relation">Heterogeneous relation</a> – Relationship between elements of two sets<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li> <li><a href="/wiki/Reflexive_relation" title="Reflexive relation">Reflexive relation</a> – Binary relation that relates every element to itself</li> <li><a href="/wiki/Reflexive_property_of_equality" class="mw-redirect" title="Reflexive property of equality">Reflexive property of equality</a> – Basic notion of sameness in mathematics<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li> <li><a href="/wiki/Symmetric_relation" title="Symmetric relation">Symmetric relation</a> – Type of binary relation</li> <li><a href="/wiki/Symmetric_property_of_equality" class="mw-redirect" title="Symmetric property of equality">Symmetric property of equality</a> – Basic notion of sameness in mathematics<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li> <li><a href="/wiki/Antisymmetric_relation" title="Antisymmetric relation">Antisymmetric relation</a> – Binary relation such that if A is related to B and is different from it then B is not related to A</li> <li><a href="/wiki/Transitive_relation" title="Transitive relation">Transitivity (mathematics)</a> – Type of binary relation <ul><li><a href="/wiki/Transitive_closure" title="Transitive closure">Transitive closure</a> – Smallest transitive relation containing a given binary relation</li> <li><a href="/wiki/Transitive_property_of_equality" class="mw-redirect" title="Transitive property of equality">Transitive property of equality</a> – Basic notion of sameness in mathematics<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li></ul></li> <li><i>Equivalence and identity</i> <ul><li><a href="/wiki/Equivalence_relation" title="Equivalence relation">Equivalence relation</a> – Mathematical concept for comparing objects</li> <li><a href="/wiki/Equivalence_class" title="Equivalence class">Equivalence class</a> – Mathematical concept</li> <li><a href="/wiki/Equality_(mathematics)" title="Equality (mathematics)">Equality (mathematics)</a> – Basic notion of sameness in mathematics <ul><li><a href="/wiki/Inequation" title="Inequation">Inequation</a> – Mathematical statement that two values are not equal</li> <li><a href="/wiki/Inequality_(mathematics)" title="Inequality (mathematics)">Inequality (mathematics)</a> – Mathematical relation expressed with < or ≤</li></ul></li> <li><a href="/wiki/Similarity_(geometry)" title="Similarity (geometry)">Similarity (geometry)</a> – Property of objects which are scaled or mirrored versions of each other</li> <li><a href="/wiki/Congruence_(geometry)" title="Congruence (geometry)">Congruence (geometry)</a> – Relationship between two figures of the same shape and size, or mirroring each other</li> <li><a href="/wiki/Equation" title="Equation">Equation</a> – Mathematical formula expressing equality</li> <li><a href="/wiki/Identity_(mathematics)" title="Identity (mathematics)">Identity (mathematics)</a> – Equation that is satisfied for all values of the variables <ul><li><a href="/wiki/Identity_element" title="Identity element">Identity element</a> – Specific element of an algebraic structure</li> <li><a href="/wiki/Identity_function" title="Identity function">Identity function</a> – In mathematics, a function that always returns the same value that was used as its argument</li></ul></li> <li><a href="/wiki/Substitution_property_of_equality" class="mw-redirect" title="Substitution property of equality">Substitution property of equality</a> – Basic notion of sameness in mathematics<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li> <li><a href="/wiki/Graphing_equivalence" class="mw-redirect" title="Graphing equivalence">Graphing equivalence</a> – Mathematical concept for comparing objects<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li> <li><a href="/wiki/Extensionality" title="Extensionality">Extensionality</a> – Logic principle</li> <li><a href="/wiki/Uniqueness_quantification" title="Uniqueness quantification">Uniqueness quantification</a> – Logical property of being the one and only object satisfying a condition</li></ul></li></ul> <div class="mw-heading mw-heading3"><h3 id="Mathematical_phraseology">Mathematical phraseology</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Outline_of_discrete_mathematics&action=edit&section=8" title="Edit section: Mathematical phraseology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/If_and_only_if" title="If and only if">If and only if</a> – Logical connective</li> <li><a href="/wiki/Necessary_and_sufficient" class="mw-redirect" title="Necessary and sufficient">Necessary and sufficient</a> – Terms to describe a conditional relationship between two statements<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li> <li><a href="/wiki/Distinct_(mathematics)" class="mw-redirect" title="Distinct (mathematics)">Distinct</a> – Basic notion of sameness in mathematics<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li> <li><a href="/wiki/Subtraction" title="Subtraction">Difference</a> – One of the four basic arithmetic operations</li> <li><a href="/wiki/Absolute_value" title="Absolute value">Absolute value</a> – Distance from zero to a number</li> <li><a href="/wiki/Up_to" title="Up to">Up to</a> – Mathematical statement of uniqueness, except for an equivalent structure (equivalence relation)</li> <li><a href="/wiki/Modular_arithmetic" title="Modular arithmetic">Modular arithmetic</a> – Computation modulo a fixed integer</li> <li><a href="/wiki/Characterization_(mathematics)" title="Characterization (mathematics)">Characterization (mathematics)</a> – Term in mathematics</li> <li><a href="/wiki/Normal_form_(mathematics)" class="mw-redirect" title="Normal form (mathematics)">Normal form</a> – Standard representation of a mathematical object<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li> <li><a href="/wiki/Canonical_form" title="Canonical form">Canonical form</a> – Standard representation of a mathematical object</li> <li><a href="/wiki/Without_loss_of_generality" title="Without loss of generality">Without loss of generality</a> – Expression in mathematics</li> <li><a href="/wiki/Vacuous_truth" title="Vacuous truth">Vacuous truth</a> – Conditional statement which is true because the antecedent cannot be satisfied</li> <li><a href="/wiki/Contradiction" title="Contradiction">Contradiction</a> – Logical incompatibility between two or more propositions, <a href="/wiki/Reductio_ad_absurdum" title="Reductio ad absurdum">Reductio ad absurdum</a> – Argument that leads to a logical absurdity</li> <li><a href="/wiki/Counterexample" title="Counterexample">Counterexample</a> – Exception to a proposed general rule</li> <li><a href="/wiki/Sufficiently_large" class="mw-redirect" title="Sufficiently large">Sufficiently large</a> – mathematical concept<span style="display:none" class="category-wikidata-fallback-annotation">Pages displaying wikidata descriptions as a fallback</span></li> <li><a href="/wiki/Pons_asinorum" title="Pons asinorum">Pons asinorum</a> – Statement that the angles opposite the equal sides of an isosceles triangle are themselves equal</li> <li><a href="/wiki/Table_of_mathematical_symbols" class="mw-redirect" title="Table of mathematical symbols">Table of mathematical symbols</a></li> <li><a href="/wiki/Contrapositive" class="mw-redirect" title="Contrapositive">Contrapositive</a> – Mathematical logic concept<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li> <li><a href="/wiki/Mathematical_induction" title="Mathematical induction">Mathematical induction</a> – Form of mathematical proof</li></ul> <div class="mw-heading mw-heading3"><h3 id="Combinatorics">Combinatorics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Outline_of_discrete_mathematics&action=edit&section=9" title="Edit section: Combinatorics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Combinatorics" title="Combinatorics">Combinatorics</a> – Branch of discrete mathematics </p> <ul><li><a href="/wiki/Combination" title="Combination">Permutations and combinations</a> – Selection of items from a set</li> <li><a href="/wiki/Permutation" title="Permutation">Permutation</a> – Mathematical version of an order change</li> <li><a href="/wiki/Combination" title="Combination">Combination</a> – Selection of items from a set</li> <li><a href="/wiki/Factorial" title="Factorial">Factorial</a> – Product of numbers from 1 to n <ul><li><a href="/wiki/Empty_product" title="Empty product">Empty product</a> – Result from multiplying no factors</li></ul></li> <li><a href="/wiki/Pascal%27s_triangle" title="Pascal's triangle">Pascal's triangle</a> – Triangular array of the binomial coefficients in mathematics</li> <li><a href="/wiki/Combinatorial_proof" title="Combinatorial proof">Combinatorial proof</a> – proofs in enumerative combinatorics based on bijections or double countings of combinatorial objects<span style="display:none" class="category-wikidata-fallback-annotation">Pages displaying wikidata descriptions as a fallback</span> <ul><li><a href="/wiki/Bijective_proof" title="Bijective proof">Bijective proof</a> – Technique for proving sets have equal size</li> <li><a href="/wiki/Double_counting_(proof_technique)" title="Double counting (proof technique)">Double counting (proof technique)</a> – Type of proof technique</li></ul></li></ul> <div class="mw-heading mw-heading3"><h3 id="Probability">Probability</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Outline_of_discrete_mathematics&action=edit&section=10" title="Edit section: Probability"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Probability" title="Probability">Probability</a> – Branch of mathematics concerning chance and uncertainty </p> <ul><li><a href="/wiki/Average" title="Average">Average</a> – Number taken as representative of a list of numbers</li> <li><a href="/wiki/Expected_value" title="Expected value">Expected value</a> – Average value of a random variable</li> <li><a href="/wiki/Discrete_random_variable" class="mw-redirect" title="Discrete random variable">Discrete random variable</a> – Variable representing a random phenomenon<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li> <li><a href="/wiki/Sample_space" title="Sample space">Sample space</a> – Set of all possible outcomes or results of a statistical trial or experiment</li> <li><a href="/wiki/Event_(probability_theory)" title="Event (probability theory)">Event</a> – In statistics and probability theory, set of outcomes to which a probability is assigned</li> <li><a href="/wiki/Conditional_Probability" class="mw-redirect" title="Conditional Probability">Conditional Probability</a> – Probability of an event occurring, given that another event has already occurred<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li> <li><a href="/wiki/Independence_(probability_theory)" title="Independence (probability theory)">Independence</a> – When the occurrence of one event does not affect the likelihood of another</li> <li><a href="/wiki/Random_variables" class="mw-redirect" title="Random variables">Random variables</a> – Variable representing a random phenomenon<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Propositional_logic">Propositional logic</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Outline_of_discrete_mathematics&action=edit&section=11" title="Edit section: Propositional logic"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Logical_operator" class="mw-redirect" title="Logical operator">Logical operator</a> – Symbol connecting sentential formulas in logic<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li> <li><a href="/wiki/Truth_table" title="Truth table">Truth table</a> – Mathematical table used in logic</li> <li><a href="/wiki/De_Morgan%27s_laws" title="De Morgan's laws">De Morgan's laws</a> – Pair of logical equivalences</li> <li><a href="/wiki/Open_sentence" class="mw-redirect" title="Open sentence">Open sentence</a> – Formula that contains at least one free variable<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li> <li><a href="/wiki/List_of_topics_in_logic" class="mw-redirect" title="List of topics in logic">List of topics in logic</a> – Overview of and topical guide to logic<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Mathematicians_associated_with_discrete_mathematics">Mathematicians associated with discrete mathematics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Outline_of_discrete_mathematics&action=edit&section=12" title="Edit section: Mathematicians associated with discrete mathematics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style 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height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Outline_of_discrete_mathematics" title="Special:EditPage/Outline of discrete mathematics">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. 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<a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Outline+of+discrete+mathematics%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">August 2024</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <ul><li><a href="/wiki/Paul_Erd%C5%91s" title="Paul Erdős">Paul Erdős</a> – Hungarian mathematician (1913–1996)</li> <li><b>Leonhard Euler - Swiss mathematician (1707-1783)</b></li> <li><b>Claude Shannon - American mathematician (1916-2001)</b></li> <li><b>Donald Knuth - American mathematician and computer scientist (b. 1938)</b></li> <li><a href="/wiki/Aristotle" title="Aristotle">Aristotle</a> – Ancient Greek philosopher and polymath (384–322 BC)</li></ul> <div class="mw-heading mw-heading2"><h2 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Johnsonbaugh</a>, <i>Discrete Mathematics</i>, Prentice Hall, 2008; James Franklin, <a rel="nofollow" class="external text" href="http://scholarship.claremont.edu/jhm/vol7/iss2/18/">Discrete and continuous: a fundamental dichotomy in mathematics</a>, <i>Journal of Humanistic Mathematics</i> 7 (2017), 355-378.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><span class="citation mathworld" id="Reference-Mathworld-Discrete_mathematics"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited 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.cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/DiscreteMathematics.html">"Discrete mathematics"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Discrete+mathematics&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FDiscreteMathematics.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOutline+of+discrete+mathematics" class="Z3988"></span></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Outline_of_discrete_mathematics&action=edit&section=15" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20180714042048/http://archives.math.utk.edu/topics/discreteMath.html">Archives</a></li> <li>Jonathan Arbib & John Dwyer, <i>Discrete Mathematics for Cryptography</i>, 1st Edition <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-907934-01-8" title="Special:BookSources/978-1-907934-01-8">978-1-907934-01-8</a>.</li> <li>John Dwyer & Suzy Jagger, <i>Discrete Mathematics for Business & Computing</i>, 1st Edition 2010 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-907934-00-1" title="Special:BookSources/978-1-907934-00-1">978-1-907934-00-1</a>.</li></ul> <div class="navbox-styles"><style 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