In mathematics, equality is a relationship between two quantities or expressions, stating that they have the same value, or represent the same mathematical object.[1] Equality between and is written, and pronounced " equals ". In this equality, and are distinguished by calling them left-hand side (LHS), and right-hand side (RHS). Two objects that are not equal are said to be distinct.
A formula such as
x=y,
1.5=3/2,
\{x\midx\in\Zand0<x\le3\}=\{1,2,3\},
The truth of an equality depends on an interpretation of its members. In the above examples, the equalities are true if the members are interpreted as numbers or sets, but are false if the members are interpreted as expressions or sequences of symbols.
An identity, such as
(x+1)2=x2+2x+1,
The word is derived from the Latin aequālis ("equal", "like", "comparable", "similar"), which itself stems from aequus ("equal", "level", "fair", "just").[6]
a>0
b>0
f(x)
f(a)=f(b)
2a-5=2b-5
f(x)=2x-5
a2=2b2
a2/b2=2
f(x,y)=x/y2
y=b
g(a)=h(a)
S
S
S
S
In logic, a predicate is a proposition which may have some free variables. Equality is a predicate, which may be true for some values of the variables (if any) and false for other values. More specifically, equality is a binary relation (i.e., a two-argument predicate) which may produce a truth value (true or false) from its arguments. In computer programming, equality is called a Boolean-valued expression, and its computation from the two expressions is known as comparison.
See also:
An equation is the problem of finding values of some variable, called, for which the specified equality is true. Each value of the unknown for which the equation holds is called a of the given equation; also stated as the equation. For example, the equation
x2-6x+5=0
x=1
x=5
An equation can be used to define a set. For example, the set of all solution pairs
(x,y)
x2+y2=1
See also: Equation solving
An identity is an equality that is true for all values of its variables in a given domain.[11] 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. An example is
\left(x+1\right)\left(x+1\right)=x2+2x+1
x
\left(x+1\right)\left(x+1\right)\equivx2+2x+1.
In mathematical logic and mathematical philosophy, equality is often described through the following properties:[14] [15] [16]
\foralla(a=a)
a
a=a
(a=b)\impliesl[\phi(a) ⇒ \phi(b)r]
\phi(x),
a=b
\phi(a)
\phi(b)
For example: For all real numbers and, if, then implies (here,
\phi(x)
These properties offer a formal reinterpretation of equality from how it is defined in standard Zermelo–Fraenkel set theory (ZFC) or other formal foundations. In ZFC, equality only means that two sets have the same elements. However, outside of set theory, mathematicians don't tend to view their objects of interest as sets. For instance, many mathematicians would say that the expression "
1\cup2
Note that this says "Equality implies these two properties" not that "These properties define equality"; this is intentional. This makes it an incomplete axiomatization of equality. That is, it does not say what equality is, only what "equality" must satify. However, the two axioms as stated are still generally useful, even as an incomplete axiomatization of equality, as they are usually sufficient for deducing most properties of equality that mathematicians care about.[17] (See the following subsection)
If these properties were to define a complete axiomatization of equality, meaning, if they were to define equality, then the converse of the second statement must be true. The converse of the Substitution property is the identity of indiscernibles, which states that two distinct things cannot have all their properties in common. In mathematics, the identity of indiscernibles is usually rejected since indiscernibles in mathematical logic are not necessarily forbidden. Set equality in ZFC is capable of declairing these indiscernibles as not equal, but an equality solely defined by these properties is not. Thus these properties form a strictly weaker notion of equality than set equality in ZFC. Outside of pure math, the identity of indiscernibles has attracted much controversy and criticism, especially from corpuscular philosophy and quantum mechanics.[18] This is why the properties are said to not form a complete axiomatization.
However, apart from cases dealing with indiscernibles, these properties taken as axioms of equality are equivalent to equality as defined in ZFC.
These are sometimes taken as the definition of equality, such as in some areas of first-order logic.[19]
xRy\Leftrightarrowx=y
a\inS
a=a
aRa
The Law of identity is distinct from reflexivity in two main ways: first, the Law of Identity applies only to cases of equality, and second, it is not restricted to elements of a set. However, many mathematicians refer to both as "Reflexivity", which is generally harmless.[20]
xRy\Leftrightarrowx=y
a,b\inS
aRb
\phi(x):xRa
(a=b)\implies(aRa ⇒ bRa)
a=b
aRa
bRa
xRy\Leftrightarrowx=y
a,b,c\inS
aRb
bRc
\phi(x):xRc
(b=a)\implies(bRc ⇒ aRc)
b=a
bRc
aRc
f(x)
\phi(x):f(a)=f(x)
(a=b)\implies[(f(a)=f(a)) ⇒ (f(a)=f(b))]
a=b
f(a)=f(a)
f(a)=f(b)
This is also sometimes included in the axioms of equality, but isn't necessary as it can be deduced from the other two axioms as shown above.
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 operations, the logarithm and the exponential function. In other words, there cannot exist any algorithm for deciding such an equality (see Richardson's theorem).
The binary relation "is approximately equal" (denoted by the symbol
≈
A questionable equality under test may be denoted using the
\stackrel{?}{=}
See main article: Equivalence relation, Isomorphism, Congruence relation and Congruence (geometry).
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 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. For example, one may distinguish fractions from rational numbers, the latter being equivalence classes of fractions: the fractions
1/2
2/4
Similarly, the sets
\{A,B,C\}
\{1,2,3\}
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
A\mapsto1,B\mapsto2,C\mapsto3.
However, there are other choices of isomorphism, such as
A\mapsto3,B\mapsto2,C\mapsto1,
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 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
See main article: Axiom of extensionality.
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.
In first-order logic with equality, the axiom of extensionality states that two sets which contain the same elements are the same set.[25]
x=y\implies\forallz,(z\inx\iffz\iny)
x=y\implies\forallz,(x\inz\iffy\inz)
(\forallz,(z\inx\iffz\iny))\impliesx=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."[26]
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.[27]
(x=y) := \forallz,(z\inx\iffz\iny)
x=y\implies\forallz,(x\inz\iffy\inz)