equality (mathematics)

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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...
, equality is a relationship between two quantities or, more generally two
mathematical expression In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

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 proofs ...
. The equality between and is written , and pronounced equals . The symbol "" is called an "
equals sign The equals sign (, ) or equal sign (), formerly known as the equality sign, is the , which is used to indicate in some sense. In an , it is placed between two that have the same value, or for which one studies the conditions under which the ...

". Two objects that are not equal are said to be distinct. For example: * $x=y$ means that and denote the same object. * The
identity Identity may refer to: Social sciences * Identity (social science), personhood or group affiliation in psychology and sociology Group expression and affiliation * Cultural identity, a person's self-affiliation (or categorization by others ...
$\left(x+1\right)^2=x^2+2x+1$ 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 Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
. * $\ = \$ if and only if $P\left(x\right) \Leftrightarrow Q\left(x\right).$ This assertion, which uses
set-builder notation In set theory illustrating the intersection (set theory), intersection of two set (mathematics), sets. Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any ...
, 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 illustrating the intersection of two sets 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 ...
, called
axiom of extensionality In and the branches of , , and that use it, the axiom of extensionality, or axiom of extension, is one of the s of . Formal statement In the of the Zermelo–Fraenkel axioms, the axiom reads: :\forall A \, \forall B \, ( \forall X \, (X \in ...
.

# 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 words. By extension, th ...
of the word is from the Latin '''' (“equal”, “like”, “comparable”, “similar”) from '''' (“equal”, “level”, “fair”, “just”).

# Basic properties

These last three properties make equality an
equivalence relation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
. They were originally included among the
Peano axioms In mathematical logic Mathematical logic, also called formal logic, is a subfield of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (alge ...
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 (linguistics), sentence. In philosophy, "Meaning (philosophy), meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same mea ...
, 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 Set (mathematics), sets and is a new set of ordered pairs consisting of elem ...
(i.e., a two-argument
predicate Predicate or predication may refer to: Computer science *Syntactic predicate (in parser technology) guidelines the parser process Linguistics *Predicate (grammar), a grammatical component of a sentence Philosophy and logic * Predication (philo ...
) which may produce a
truth value In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, ...
(''false'' or ''true'') from its arguments. In
computer programming Computer programming is the process of designing and building an executable In computing, executable code, an executable file, or an executable program, sometimes simply referred to as an executable or binary, causes a computer "to perform in ...
, its computation from the two expressions is known as
comparison File:Comparison of dietary fat composition.png, A chart showing a comparison of qualities of a variety of cooking oils, aimed at helping the reader decide which choices would be best for their health. Comparison or comparing is the act of evaluat ...
.

# Identities

When ''A'' and ''B'' may be viewed as
functions Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
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 Identity may refer to: Social sciences * Identity (social science), personhood or group affiliation in psychology and sociology Group expression and affiliation * Cultural identity, a person's self-affiliation (or categorization by others ...
. 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 featuring the equal sign The equals sign (British English, Unicode Consortium) or equal sign (American English), ...
: $\left\left(x + 1\right\right)\left\left(x + 1\right\right) \equiv x^2 + 2 x + 1.$

# Equations

An equation 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. 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.

# Congruences

In some cases, one may consider as equal two mathematical objects that are only equivalent for the properties being considered. In geometry for instance, two geometric shapes are said to be equal when one may be moved to coincide with the other. The word congruence (and the associated symbol $\cong$) is also used for this kind of equality.

# Approximate equality

There are some mathematical logic, logic systems that do not have any notion of equality. This reflects the undecidable problem, undecidability of the equality of two real numbers, defined by formulas involving the integers, the basic arithmetic operations, the logarithm and the exponential function. In other words, there cannot exist any algorithm 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 Set (mathematics), sets and is a new set of ordered pairs consisting of elem ...
"approximation, is approximately equal" (denoted by the symbol $\approx$) between real numbers or other things, even if more precisely defined, is not transitive (since many small Difference (mathematics), differences can add up to something big). However, equality almost everywhere ''is'' transitive. A questionable equality under test may be denoted using the ≟ symbol.

# Relation with equivalence and isomorphism

Viewed as a relation, equality is the archetype of the more general concept of an
equivalence relation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
on a set: those binary relations that are reflexive relation, reflexive, symmetric relation, symmetric and transitive relation, 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 relation, equivalence'' or ''isomorphism.'' For example, one may distinguish ''fraction (mathematics), fractions'' from ''rational numbers,'' 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 $\$ are not equal sets — the first consists of letters, while the second consists of numbers — but they are both sets of three elements and thus isomorphic, meaning that there is a bijection between them. For example :$\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, Isomorphism#Relation with equality, between equality and isomorphism, is of fundamental importance in category theory and is one motivation for the development of category theory.

# Logical definitions

Leibniz characterized the notion of equality as follows: : Given any ''x'' and ''y'', ''x'' = ''y'' if and only if, given any Predicate (mathematics), 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'')