The following outline is presented as an overview of and topical guide to discrete mathematics:

Discrete mathematics

Discrete mathematics

Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not...

– study of mathematical

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 structures

Mathematical structure

In mathematics, a structure on a set, or more generally a type, consists of additional mathematical objects that in some manner attach to the set, making it easier to visualize or work with, or endowing the collection with meaning or significance....

 that are fundamentally discrete

Discrete space

In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are "isolated" from each other in a certain sense.- Definitions :Given a set X:...

 rather than continuous

Continuous function

In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

. In contrast to real number

Real number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

s that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integer

Integer

The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

s, graphs

Graph (mathematics)

In mathematics, a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges...

, and statements in logic

Mathematical logic

Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...

 – do not vary smoothly in this way, but have distinct, separated values. Discrete mathematics therefore excludes topics in "continuous mathematics" such as calculus

Calculus

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

 and analysis

Mathematical analysis

Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...

.

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 terms of art that may be encountered.

Subjects in discrete mathematics

• Logic
  Logic
  In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
  
   – a study of reasoning
• Set theory
  Set theory
  Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
  
   – a study of collections of elements
• Number theory
  Number theory
  Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
  
   –
• Combinatorics
  Combinatorics
  Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
  
   – a study of counting
• Graph theory
  Graph theory
  In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
  
   –
• Digital geometry
  Digital geometry
  Digital geometry deals with discrete sets considered to be digitized models or images of objects of the 2D or 3D Euclidean space.Simply put, digitizing is replacing an object by a discrete set of its points...
  
   and digital topology
  Digital topology
  Digital topology deals with properties and features of two-dimensional or three-dimensional digital imagesthat correspond to topological properties or topological features of objects....
  
  
• Algorithmics – a study of methods of calculation
• Information theory
  Information theory
  Information theory is a branch of applied mathematics and electrical engineering involving the quantification of information. Information theory was developed by Claude E. Shannon to find fundamental limits on signal processing operations such as compressing data and on reliably storing and...
  
   –
• Computability and complexity
  Complexity
  In general usage, complexity tends to be used to characterize something with many parts in intricate arrangement. The study of these complex linkages is the main goal of complex systems theory. In science there are at this time a number of approaches to characterizing complexity, many of which are...
  
   theories – dealing with theoretical and practical limitations of algorithms
• Elementary probability theory
  Probability theory
  Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
  
   and Markov chain
  Markov chain
  A Markov chain, named after Andrey Markov, is a mathematical system that undergoes transitions from one state to another, between a finite or countable number of possible states. It is a random process characterized as memoryless: the next state depends only on the current state and not on the...
  
  s
• Linear algebra
  Linear algebra
  Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
  
   – a study of related linear equations
• Functions
  Function (mathematics)
  In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
  
   –
• Partially ordered set
  Partially ordered set
  In mathematics, especially order theory, a partially ordered set formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the...
  
   –
• Probability
  Probability
  Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
  
   –
• Proofs
  Mathematical proof
  In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...
  
   –
• Counting
  Counting
  Counting is the action of finding the number of elements of a finite set of objects. The traditional way of counting consists of continually increasing a counter by a unit for every element of the set, in some order, while marking those elements to avoid visiting the same element more than once,...
  
   –
• Relation
  Relation (mathematics)
  In set theory and logic, a relation is a property that assigns truth values to k-tuples of individuals. Typically, the property describes a possible connection between the components of a k-tuple...
  
   –

Discrete mathematical disciplines

For further reading in discrete mathematics, beyond a basic level, see these pages. Many of these disciplines are closely related to computer science

Computer science

Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...

.

• Automata theory
  Automata theory
  In theoretical computer science, automata theory is the study of abstract machines and the computational problems that can be solved using these machines. These abstract machines are called automata...
  
   –
• Combinatorics
  Combinatorics
  Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
  
    –
• Combinatorial geometry  –
• Computational geometry
  Computational geometry
  Computational geometry is a branch of computer science devoted to the study of algorithms which can be stated in terms of geometry. Some purely geometrical problems arise out of the study of computational geometric algorithms, and such problems are also considered to be part of computational...
  
    –
• Digital geometry
  Digital geometry
  Digital geometry deals with discrete sets considered to be digitized models or images of objects of the 2D or 3D Euclidean space.Simply put, digitizing is replacing an object by a discrete set of its points...
  
    –
• Discrete geometry
  Discrete geometry
  Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles,...
  
    –
• Graph theory
  Graph theory
  In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
  
   –
• Mathematical logic
  Mathematical logic
  Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
  
   –
• Combinatorial optimization
  Optimization (mathematics)
  In mathematics, computational science, or management science, mathematical optimization refers to the selection of a best element from some set of available alternatives....
  
    –
• Set theory
  Set theory
  Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
  
   –
• Combinatorial topology
  Topology
  Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
  
   –
• Number theory
  Number theory
  Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
  
   –
• Information theory
  Information theory
  Information theory is a branch of applied mathematics and electrical engineering involving the quantification of information. Information theory was developed by Claude E. Shannon to find fundamental limits on signal processing operations such as compressing data and on reliably storing and...
  
   –
• Game theory
  Game theory
  Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...
  
   –

Sets

• Set (mathematics) –
  • Element (mathematics) –
  • Venn diagram
    Venn diagram
    Venn diagrams or set diagrams are diagrams that show all possible logical relations between a finite collection of sets . Venn diagrams were conceived around 1880 by John Venn...
    
     –
  • Empty set
    Empty set
    In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
    
     –
  • Subset
    Subset
    In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
    
     –
  • Union (set theory)
    Union (set theory)
    In set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets S_1, S_2, S_3, \dots , S_n\,\! gives a set S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n.- Definition :...
    
     –
    • Disjoint union
      Disjoint union
      In mathematics, the term disjoint union may refer to one of two different concepts:* In set theory, a disjoint union is a modified union operation that indexes the elements according to which set they originated in; disjoint sets have no element in common.* In probability theory , a disjoint union...
      
       –
  • Intersection (set theory)
    Intersection (set theory)
    In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....
    
     –
    • Disjoint sets –
  • Complement (set theory)
    Complement (set theory)
    In set theory, a complement of a set A refers to things not in , A. The relative complement of A with respect to a set B, is the set of elements in B but not in A...
    
     –
  • Symmetric difference
    Symmetric difference
    In mathematics, the symmetric difference of two sets is the set of elements which are in either of the sets and not in their intersection. The symmetric difference of the sets A and B is commonly denoted by A\,\Delta\,B\,orA \ominus B....
    
     –
• Ordered pair
  Ordered pair
  In mathematics, an ordered pair is a pair of mathematical objects. In the ordered pair , the object a is called the first entry, and the object b the second entry of the pair...
  
   –
• Cartesian product
  Cartesian product
  In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
  
   –
• Power set –
• Simple theorems in the algebra of sets
  Simple theorems in the algebra of sets
  The simple theorems in the algebra of sets are some of the elementary properties of the algebra of union , intersection , and set complement of sets....
  
   –
• Naive set theory
  Naive set theory
  Naive set theory is one of several theories of sets used in the discussion of the foundations of mathematics. The informal content of this naive set theory supports both the aspects of mathematical sets familiar in discrete mathematics , and the everyday usage of set theory concepts in most...
  
   –
• Multiset
  Multiset
  In mathematics, the notion of multiset is a generalization of the notion of set in which members are allowed to appear more than once...
  
   –

Functions

• Function
  Function (mathematics)
  In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
  
   –
• Domain of a function –
• Codomain
  Codomain
  In mathematics, the codomain or target set of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation...
  
   –
• Range of a function –
• Image (mathematics)
  Image (mathematics)
  In mathematics, an image is the subset of a function's codomain which is the output of the function on a subset of its domain. Precisely, evaluating the function at each element of a subset X of the domain produces a set called the image of X under or through the function...
  
   –
• Injective function
  Injective function
  In mathematics, an injective function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain. In other words, every element of the function's codomain is mapped to by at most one element of its domain...
  
   –
• Surjection –
• Bijection
  Bijection
  A bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...
  
   –
• Function composition
  Function composition
  In mathematics, function composition is the application of one function to the results of another. For instance, the functions and can be composed by computing the output of g when it has an argument of f instead of x...
  
   –
• Partial function
  Partial function
  In mathematics, a partial function from X to Y is a function ƒ: X' → Y, where X' is a subset of X. It generalizes the concept of a function by not forcing f to map every element of X to an element of Y . If X' = X, then ƒ is called a total function and is equivalent to a function...
  
   –
• Multivalued function
  Multivalued function
  In mathematics, a multivalued function is a left-total relation; i.e. every input is associated with one or more outputs...
  
   –
• Binary function
  Binary function
  In mathematics, a binary function, or function of two variables, is a function which takes two inputs.Precisely stated, a function f is binary if there exists sets X, Y, Z such that\,f \colon X \times Y \rightarrow Z...
  
   –
• Floor function
  Floor function
  In mathematics and computer science, the floor and ceiling functions map a real number to the largest previous or the smallest following integer, respectively...
  
   –
• Sign function
  Sign function
  In mathematics, the sign function is an odd mathematical function that extracts the sign of a real number. To avoid confusion with the sine function, this function is often called the signum function ....
  
   –
• Inclusion map
  Inclusion map
  In mathematics, if A is a subset of B, then the inclusion map is the function i that sends each element, x of A to x, treated as an element of B:i: A\rightarrow B, \qquad i=x....
  
   –
• Pigeonhole principle –
• Relation composition –
• Permutations –
• Symmetry
  Symmetry
  Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...
  
   –

Operations

Binary operator –

• Associativity
  Associativity
  In mathematics, associativity is a property of some binary operations. It means that, within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not...
  
   –
• Commutativity
  Commutativity
  In mathematics an operation is commutative if changing the order of the operands does not change the end result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it...
  
   –
• Distributivity
  Distributivity
  In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalizes the distributive law from elementary algebra.For example:...
  
  

Arithmetic

Decimal

Decimal

The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....

 –

• Binary numeral system
  Binary numeral system
  The binary numeral system, or base-2 number system, represents numeric values using two symbols, 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2...
  
   –
• Divisor
  Divisor
  In mathematics, a divisor of an integer n, also called a factor of n, is an integer which divides n without leaving a remainder.-Explanation:...
  
   –
• Division by zero
  Division by zero
  In mathematics, division by zero is division where the divisor is zero. Such a division can be formally expressed as a / 0 where a is the dividend . Whether this expression can be assigned a well-defined value depends upon the mathematical setting...
  
   –
• Indeterminate form
  Indeterminate form
  In calculus and other branches of mathematical analysis, an indeterminate form is an algebraic expression obtained in the context of limits. Limits involving algebraic operations are often performed by replacing subexpressions by their limits; if the expression obtained after this substitution...
  
   –
• Empty product
  Empty product
  In mathematics, an empty product, or nullary product, is the result of multiplying no factors. It is equal to the multiplicative identity 1, given that it exists for the multiplication operation in question, just as the empty sum—the result of adding no numbers—is zero, or the additive...
  
   –
• Euclidean algorithm
  Euclidean algorithm
  In mathematics, the Euclidean algorithm is an efficient method for computing the greatest common divisor of two integers, also known as the greatest common factor or highest common factor...
  
   –
• Fundamental theorem of arithmetic
  Fundamental theorem of arithmetic
  In number theory, the fundamental theorem of arithmetic states that any integer greater than 1 can be written as a unique product of prime numbers...
  
   –
• Modular arithmetic
  Modular arithmetic
  In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
  
   –
• Successor function

Elementary algebra

Left-hand side and right-hand side of an equation –

• Linear equation
  Linear equation
  A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable....
  
   –
• Quadratic equation
  Quadratic equation
  In mathematics, a quadratic equation is a univariate polynomial equation of the second degree. A general quadratic equation can be written in the formax^2+bx+c=0,\,...
  
   –
• Solution point –
• Arithmetic progression
  Arithmetic progression
  In mathematics, an arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant...
  
   –
• Recurrence relation
  Recurrence relation
  In mathematics, a recurrence relation is an equation that recursively defines a sequence, once one or more initial terms are given: each further term of the sequence is defined as a function of the preceding terms....
  
   –
• Finite difference
  Finite difference
  A finite difference is a mathematical expression of the form f − f. If a finite difference is divided by b − a, one gets a difference quotient...
  
   –
• Difference operator –
• Groups
  Group (mathematics)
  In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
  
   –
• Group isomorphism
  Group isomorphism
  In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two groups, then the groups are called isomorphic...
  
   –
• Subgroups –
• Fermat's little theorem
  Fermat's little theorem
  Fermat's little theorem states that if p is a prime number, then for any integer a, a p − a will be evenly divisible by p...
  
   –
• Cryptography
  Cryptography
  Cryptography is the practice and study of techniques for secure communication in the presence of third parties...
  
   –
• Faulhaber's formula –

Mathematical relations

• Binary relation
  Binary relation
  In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = . More generally, a binary relation between two sets A and B is a subset of...
  
   –
• Mathematical relation
  Relation (mathematics)
  In set theory and logic, a relation is a property that assigns truth values to k-tuples of individuals. Typically, the property describes a possible connection between the components of a k-tuple...
  
   –
• Reflexive relation
  Reflexive relation
  In mathematics, a reflexive relation is a binary relation on a set for which every element is related to itself, i.e., a relation ~ on S where x~x holds true for every x in S. For example, ~ could be "is equal to".-Related terms:...
  
   –
• Reflexive property of equality –
• Symmetric relation
  Symmetric relation
  In mathematics, a binary relation R over a set X is symmetric if it holds for all a and b in X that if a is related to b then b is related to a.In mathematical notation, this is:...
  
   –
• Symmetric property of equality –
• Antisymmetric relation
  Antisymmetric relation
  In mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in Xor, equivalently,In mathematical notation, this is:\forall a, b \in X,\ R \and R \; \Rightarrow \; a = bor, equivalently,...
  
   –
• Transitivity (mathematics)
  Transitive relation
  In mathematics, a binary relation R over a set X is transitive if whenever an element a is related to an element b, and b is in turn related to an element c, then a is also related to c....
  
   –
  • Transitive closure
    Transitive closure
    In mathematics, the transitive closure of a binary relation R on a set X is the transitive relation R+ on set X such that R+ contains R and R+ is minimal . If the binary relation itself is transitive, then the transitive closure will be that same binary relation; otherwise, the transitive closure...
    
     –
  • Transitive property of equality –
• Equivalence and identity
  • Equivalence relation
    Equivalence relation
    In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...
    
     –
  • Equivalence class –
  • Equality (mathematics) –
    • Inequation
      Inequation
      In mathematics, an inequation is a statement that two objects or expressions are not the same, or do not represent the same value. This relation is written with a crossed-out equal sign as inx \neq y....
      
       –
    • Inequality (mathematics) –
  • Similarity (geometry)
    Similarity (geometry)
    Two geometrical objects are called similar if they both have the same shape. More precisely, either one is congruent to the result of a uniform scaling of the other...
    
     –
  • Congruence (geometry)
    Congruence (geometry)
    In geometry, two figures are congruent if they have the same shape and size. This means that either object can be repositioned so as to coincide precisely with the other object...
    
     –
  • Equation
    Equation
    An equation is a mathematical statement that asserts the equality of two expressions. In modern notation, this is written by placing the expressions on either side of an equals sign , for examplex + 3 = 5\,asserts that x+3 is equal to 5...
    
     –
  • Identity (mathematics)
    Identity (mathematics)
    In mathematics, the term identity has several different important meanings:*An identity is a relation which is tautologically true. This means that whatever the number or value may be, the answer stays the same. For example, algebraically, this occurs if an equation is satisfied for all values of...
    
     –
    • Identity element
      Identity element
      In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...
      
       –
    • Identity function
      Identity function
      In mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that was used as its argument...
      
       –
  • Substitution property of equality –
  • Graphing equivalence –
  • Extensionality
    Extensionality
    In logic, extensionality, or extensional equality refers to principles that judge objects to be equal if they have the same external properties...
    
     –
  • Uniqueness quantification
    Uniqueness quantification
    In mathematics and logic, the phrase "there is one and only one" is used to indicate that exactly one object with a certain property exists. In mathematical logic, this sort of quantification is known as uniqueness quantification or unique existential quantification.Uniqueness quantification is...
    
     –

Mathematical phraseology

If and only if

If and only if

In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....

  –

• Necessary and sufficient (Sufficient condition) –
• Distinct –
• Difference
  Subtraction
  In arithmetic, subtraction is one of the four basic binary operations; it is the inverse of addition, meaning that if we start with any number and add any number and then subtract the same number we added, we return to the number we started with...
  
   –
• Absolute value
  Absolute value
  In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...
  
   –
• Up to
  Up to
  In mathematics, the phrase "up to x" means "disregarding a possible difference in  x".For instance, when calculating an indefinite integral, one could say that the solution is f "up to addition by a constant," meaning it differs from f, if at all, only by some constant.It indicates that...
  
   –
• Modular arithmetic
  Modular arithmetic
  In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
  
   –
• Characterization (mathematics)
  Characterization (mathematics)
  In mathematics, the statement that "Property P characterizes object X" means, not simply that X has property P, but that X is the only thing that has property P. It is also common to find statements such as "Property Q characterises Y up to isomorphism". The first type of statement says in...
  
   –
• Normal form –
• Canonical form
  Canonical form
  Generally, in mathematics, a canonical form of an object is a standard way of presenting that object....
  
   –
• Without loss of generality
  Without loss of generality
  Without loss of generality is a frequently used expression in mathematics...
  
   –
• Vacuous truth
  Vacuous truth
  A vacuous truth is a truth that is devoid of content because it asserts something about all members of a class that is empty or because it says "If A then B" when in fact A is inherently false. For example, the statement "all cell phones in the room are turned off" may be true...
  
   –
• Contradiction
  Contradiction
  In classical logic, a contradiction consists of a logical incompatibility between two or more propositions. It occurs when the propositions, taken together, yield two conclusions which form the logical, usually opposite inversions of each other...
  
  , Reductio ad absurdum
  Reductio ad absurdum
  In logic, proof by contradiction is a form of proof that establishes the truth or validity of a proposition by showing that the proposition's being false would imply a contradiction...
  
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• Counterexample
  Counterexample
  In logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule. For example, consider the proposition "all students are lazy"....
  
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• Sufficiently large –
• Pons asinorum
  Pons asinorum
  Pons asinorum is the name given to Euclid's fifth proposition in Book 1 of his Elements of geometry, also known as the theorem on isosceles triangles. It states that the angles opposite the equal sides of an isosceles triangle are equal...
  
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• Table of mathematical symbols
  Table of mathematical symbols
  This is a listing of common symbols found within all branches of mathematics. Each symbol is listed in both HTML, which depends on appropriate fonts being installed, and in , as an image.-Symbols:-Variations:...
  
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• Contrapositive –
• Mathematical induction
  Mathematical induction
  Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers...
  
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Combinatorics

• Permutations and combinations –
• Permutation
  Permutation
  In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...
  
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• Combination
  Combination
  In mathematics a combination is a way of selecting several things out of a larger group, where order does not matter. In smaller cases it is possible to count the number of combinations...
  
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• Factorial
  Factorial
  In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n...
  
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  • Empty product
    Empty product
    In mathematics, an empty product, or nullary product, is the result of multiplying no factors. It is equal to the multiplicative identity 1, given that it exists for the multiplication operation in question, just as the empty sum—the result of adding no numbers—is zero, or the additive...
    
     –
• Pascal's triangle
  Pascal's triangle
  In mathematics, Pascal's triangle is a triangular array of the binomial coefficients in a triangle. It is named after the French mathematician, Blaise Pascal...
  
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• Combinatorial proof
  Combinatorial proof
  In mathematics, the term combinatorial proof is often used to mean either of two types of proof of an identity in enumerative combinatorics that either states that two sets of combinatorial configurations, depending on one or more parameters, have the same number of elements , or gives a formula...
  
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  • Bijective proof
    Bijective proof
    In combinatorics, bijective proof is a proof technique that finds a bijective function f : A → B between two sets A and B, thus proving that they have the same number of elements, |A| = |B|. One place the technique is useful is where we wish to know the size of A, but can find no direct way of...
    
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  • Double counting (proof technique)
    Double counting (proof technique)
    In combinatorics, double counting, also called counting in two ways, is a combinatorial proof technique for showing that two expressions are equal by demonstrating that they are two ways of counting the size of one set...
    
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Probability

• Average
  Average
  In mathematics, an average, or central tendency of a data set is a measure of the "middle" value of the data set. Average is one form of central tendency. Not all central tendencies should be considered definitions of average....
  
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• Expected value
  Expected value
  In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
  
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• Discrete random variable –
• Sample space –
• Event
  Event (probability theory)
  In probability theory, an event is a set of outcomes to which a probability is assigned. Typically, when the sample space is finite, any subset of the sample space is an event...
  
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• Conditional Probability
  Conditional probability
  In probability theory, the "conditional probability of A given B" is the probability of A if B is known to occur. It is commonly notated P, and sometimes P_B. P can be visualised as the probability of event A when the sample space is restricted to event B...
  
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• Independence –
• Random variables –

Propositional logic

Logical operator –

• Truth table
  Truth table
  A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—to compute the functional values of logical expressions on each of their functional arguments, that is, on each combination of values taken by their...
  
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• De Morgan's laws –
• Open sentence
  Open sentence
  In mathematics, an open sentence is described as "open" in the sense that its truth value is meaningless until its variables are replaced with specific numbers, at which point the truth value can usually be determined...
  
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• List of topics in logic –

External links

• Archives
• Jonathan Arbib & John Dwyer, Discrete Mathematics for Cryptography, 1st Edition ISBN 978-1907934018.
• John Dwyer & Suzy Jagger, Discrete Mathematics for Business & Computing, 1st Edition 2010 ISBN 978-1907934001.