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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, equality is a relationship between two quantities or, more generally two
mathematical expression In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context. Mathematical symbols can designate numbers ( constants), variables, operations, ...
s, asserting that the quantities have the same value, or that the expressions represent the same
mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deductive reasoning and mathematical ...
. The equality between and is written , and pronounced equals . The symbol "" is called an "
equals sign The equals sign (British English, Unicode) or equal sign (American English), also known as the equality sign, is the mathematical symbol , which is used to indicate equality in some well-defined sense. In an equation, it is placed between t ...
". Two objects that are not equal are said to be distinct. For example: * $x=y$ means that and denote the same object. * The identity $\left(x+1\right)^2=x^2+2x+1$ means that if is any number, then the two expressions have the same value. This may also be interpreted as saying that the two sides of the equals sign represent the same function. * $\ = \$ if and only if $P\left(x\right) \Leftrightarrow Q\left(x\right).$ This assertion, which uses
set-builder notation In set theory and its applications to logic, mathematics, and computer science, set-builder notation is a mathematical notation for describing a set by enumerating its elements, or stating the properties that its members must satisfy. Definin ...
, means that if the elements satisfying the property $P\left(x\right)$ are the same as the elements satisfying $Q\left(x\right),$ then the two uses of the set-builder notation define the same set. This property is often expressed as "two sets that have the same elements are equal." It is one of the usual axioms of
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
, called
axiom of extensionality In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of extensionality, or axiom of extension, is one of the axioms of Zermelo–Fraenkel set theory. It says that sets having the same elemen ...
.

# Etymology

The
etymology Etymology () The New Oxford Dictionary of English (1998) – p. 633 "Etymology /ˌɛtɪˈmɒlədʒi/ the study of the class in words and the way their meanings have changed throughout time". is the study of the history of the form of words a ...
of the word is from the Latin '' aequālis'' (“equal”, “like”, “comparable”, “similar”) from '' aequus'' (“equal”, “level”, “fair”, “just”).

# Basic properties

These last three properties make equality an
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relati ...
. They were originally included among the
Peano axioms In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly ...
for natural numbers. Although the symmetric and transitive properties are often seen as fundamental, they can be deduced from substitution and reflexive properties.

# Equality as predicate

When ''A'' and ''B'' are not fully specified or depend on some variables, equality is a
proposition In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...
, which may be true for some values and false for other values. Equality is a
binary relation In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over sets and is a new set of ordered pairs consisting of elements in and i ...
(i.e., a two-argument predicate) which may produce a
truth value In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values (''true'' or '' false''). Computing In some progra ...
(''false'' or ''true'') from its arguments. In
computer programming Computer programming is the process of performing a particular computation (or more generally, accomplishing a specific computing result), usually by designing and building an executable computer program. Programming involves tasks such as ana ...
, its computation from the two expressions is known as
comparison Comparison or comparing is the act of evaluating two or more things by determining the relevant, comparable characteristics of each thing, and then determining which characteristics of each are similar to the other, which are different, and t ...
.

# Identities

When ''A'' and ''B'' may be viewed as functions of some variables, then ''A'' = ''B'' means that ''A'' and ''B'' define the same function. Such an equality of functions is sometimes called an identity. An example is $\left\left(x + 1\right\right)\left\left(x + 1\right\right) = x^2 + 2 x + 1.$ Sometimes, but not always, an identity is written with a
triple bar The triple bar, or tribar ≡, is a symbol with multiple, context-dependent meanings. It has the appearance of an equals sign  sign with a third line. The triple bar character in Unicode is code point .. The closely related code point ...
: $\left\left(x + 1\right\right)\left\left(x + 1\right\right) \equiv x^2 + 2 x + 1.$

# Equations

An
equation In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in ...
is a problem of finding values of some variables, called , for which the specified equality is true. The term "equation" may also refer to an equality relation that is satisfied only for the values of the variables that one is interested in. For example, $x^2 + y^2 = 1$ is the of the
unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucl ...
. There is no standard notation that distinguishes an equation from an identity, or other use of the equality relation: one has to guess an appropriate interpretation from the semantics of expressions and the context. An identity is to be true for all values of variables in a given domain. An "equation" may sometimes mean an identity, but more often than not, it a subset of the variable space to be the subset where the equation is true.

# Approximate equality

There are some logic systems that do not have any notion of equality. This reflects the undecidability of the equality of two
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s, defined by formulas involving the
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s, the basic arithmetic operations, the
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of ...
and the
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, a ...
. In other words, there cannot exist any
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
for deciding such an equality. The
binary relation In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over sets and is a new set of ordered pairs consisting of elements in and i ...
" is approximately equal" (denoted by the symbol $\approx$) between
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s or other things, even if more precisely defined, is not transitive (since many small differences can add up to something big). However, equality
almost everywhere In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion t ...
''is'' transitive. A questionable equality under test may be denoted using the symbol.

# Relation with equivalence, congruence, and isomorphism

Viewed as a relation, equality is the archetype of the more general concept of an
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relati ...
on a set: those binary relations that are reflexive,
symmetric Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
and transitive. The identity relation is an equivalence relation. Conversely, let ''R'' be an equivalence relation, and let us denote by ''xR'' the equivalence class of ''x'', consisting of all elements ''z'' such that ''x R z''. Then the relation ''x R y'' is equivalent with the equality ''xR'' = ''yR''. It follows that equality is the finest equivalence relation on any set ''S'' in the sense that it is the relation that has the smallest equivalence classes (every class is reduced to a single element). In some contexts, equality is sharply distinguished from '' equivalence'' or ''
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
.'' For example, one may distinguish ''
fractions A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
'' from ''
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s,'' the latter being equivalence classes of fractions: the fractions $1/2$ and $2/4$ are distinct as fractions (as different strings of symbols) but they "represent" the same rational number (the same point on a number line). This distinction gives rise to the notion of a quotient set. Similarly, the sets :$\$ and $\$ are not equal sets — the first consists of letters, while the second consists of numbers — but they are both sets of three elements and thus isomorphic, meaning that there is a
bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
between them. For example :$\text \mapsto 1, \text \mapsto 2, \text \mapsto 3.$ However, there are other choices of isomorphism, such as :$\text \mapsto 3, \text \mapsto 2, \text \mapsto 1,$ and these sets cannot be identified without making such a choice — any statement that identifies them "depends on choice of identification". This distinction, between equality and isomorphism, is of fundamental importance in
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
and is one motivation for the development of category theory. In some cases, one may consider as equal two mathematical objects that are only equivalent for the properties and structure being considered. The word congruence (and the associated symbol $\cong$) is frequently used for this kind of equality, and is defined as the quotient set of the isomorphism classes between the objects. In
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
for instance, two
geometric shape A shape or figure is a graphical representation of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material type. A plane shape or plane figure is constrained to lie ...
s are said to be equal or congruent when one may be moved to coincide with the other, and the equality/congruence relation is the isomorphism classes of isometries between shapes. Similarly to isomorphisms of sets, the difference between isomorphisms and equality/congruence between such mathematical objects with properties and structure was one motivation for the development of
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
, as well as for homotopy type theory and univalent foundations.

# Logical definitions

Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathe ...
characterized the notion of equality as follows: : Given any ''x'' and ''y'', ''x'' = ''y''
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bico ...
, given any predicate ''P'', ''P''(''x'') if and only if ''P''(''y'').

# Equality in set theory

Equality of sets is axiomatized in set theory in two different ways, depending on whether the axioms are based on a first-order language with or without equality.

## Set equality based on first-order logic with equality

In first-order logic with equality, the axiom of extensionality states that two sets which ''contain'' the same elements are the same set. * Logic axiom: ''x'' = ''y'' ⇒ ∀''z'', (''z'' ∈ ''x'' ⇔ ''z'' ∈ ''y'') * Logic axiom: ''x'' = ''y'' ⇒ ∀''z'', (''x'' ∈ ''z'' ⇔ ''y'' ∈ ''z'') * Set theory axiom: (∀''z'', (''z'' ∈ ''x'' ⇔ ''z'' ∈ ''y'')) ⇒ ''x'' = ''y'' Incorporating half of the work into the first-order logic may be regarded as a mere matter of convenience, as noted by Lévy. : "The reason why we take up first-order predicate calculus ''with equality'' is a matter of convenience; by this we save the labor of defining equality and proving all its properties; this burden is now assumed by the logic."

## Set equality based on first-order logic without equality

In first-order logic without equality, two sets are ''defined'' to be equal if they contain the same elements. Then the axiom of extensionality states that two equal sets ''are contained in'' the same sets.. * Set theory definition: "''x'' = ''y''" means ∀''z'', (''z'' ∈ ''x'' ⇔ ''z'' ∈ ''y'') * Set theory axiom: ''x'' = ''y'' ⇒ ∀''z'', (''x'' ∈ ''z'' ⇔ ''y'' ∈ ''z'')

*
Extensionality In logic, extensionality, or extensional equality, refers to principles that judge objects to be equal if they have the same external properties. It stands in contrast to the concept of intensionality, which is concerned with whether the internal ...
* Homotopy type theory *
Inequality Inequality may refer to: Economics * Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy * Economic inequality, difference in economic well-being between population groups * ...
*
List of mathematical symbols A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula. ...
*
Logical equality Logical equality is a logical operator that corresponds to equality in Boolean algebra and to the logical biconditional in propositional calculus. It gives the functional value '' true'' if both functional arguments have the same logical v ...
*
Proportionality (mathematics) In mathematics, two sequences of numbers, often experimental data, are proportional or directly proportional if their corresponding elements have a constant ratio, which is called the coefficient of proportionality or proportionality constan ...

* * * * * * *