In
mathematics, equality is a relationship between two quantities or, more generally two
mathematical expressions, asserting that the quantities have the same value, or that the expressions represent the same
mathematical object. The equality between and is written , and pronounced equals . The symbol "" is called an "
equals sign". Two objects that are not equal are said to be distinct.
For example:
*
means that and denote the same object.
* The
identity
Identity may refer to:
* Identity document
* Identity (philosophy)
* Identity (social science)
* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film), an ...
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 type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
.
*
if and only if
This assertion, which uses
set-builder notation, means that if the elements satisfying the property
are the same as the elements satisfying
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, 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 eleme ...
.
Etymology
The
etymology 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. They were originally included among the
Peano axioms 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
Predicate or predication may refer to:
* Predicate (grammar), in linguistics
* Predication (philosophy)
* several closely related uses in mathematics and formal logic:
**Predicate (mathematical logic)
**Propositional function
**Finitary relation, o ...
) which may produce a
truth value (''false'' or ''true'') from its arguments. In
computer programming, 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
Identity may refer to:
* Identity document
* Identity (philosophy)
* Identity (social science)
* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film), an ...
. An example is
Sometimes, but not always, an identity is written with a
triple bar:
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,
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.
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 numbers, defined by formulas involving the
integers, the basic
arithmetic operation
Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers—addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th c ...
s, the
logarithm 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
) between
real numbers 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 to ...
''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 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 definit ...
and
transitive. The identity relation is an equivalence relation. Conversely, let ''R'' be an equivalence relation, and let us denote by ''x
R'' 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 ''x
R'' = ''y
R''. 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.'' For example, one may distinguish ''
fractions'' from ''
rational numbers,'' the latter being equivalence classes of fractions: the fractions
and
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
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
.
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
:
However, there are other choices of isomorphism, such as
:
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, cate ...
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
) is frequently used for this kind of equality, and is defined as the
quotient set
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
of the
isomorphism class
In mathematics, an isomorphism class is a collection of mathematical objects isomorphic to each other.
Isomorphism classes are often defined as the exact identity of the elements of the set is considered irrelevant, and the properties of the str ...
es between the objects. In
geometry for instance, two
geometric shapes 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, cate ...
, 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 mathema ...
characterized the notion of equality as follows:
:
Given any ''x'' and ''y'', ''x'' = ''y''
if and only if, given any
predicate
Predicate or predication may refer to:
* Predicate (grammar), in linguistics
* Predication (philosophy)
* several closely related uses in mathematics and formal logic:
**Predicate (mathematical logic)
**Propositional function
**Finitary relation, o ...
''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'')
See also
*
Extensionality
*
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
*
Logical equality
*
Proportionality (mathematics)
Notes
References
*
*
*
*
*
*
*
External links
*
{{DEFAULTSORT:Equality (Mathematics)
Mathematical logic
Binary relations
Elementary arithmetic
Equivalence (mathematics)