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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 A and B is written A = B.
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 called an identity. For example (x + 1)2 = x2 + 2x + 1.
In mathematical logic, equality is formalized with axioms (e.g. the first few Peano axioms, or the axiom of extensionality in ZF set theory). It can also be viewed as a relation: the identity relation, or diagonal relation, the binary relation on a set X defined by
The identity relation is the archetype of the more general concept of an equivalence relation on a set: those binary relations which are reflexive, symmetric, and transitive. The identity relation is an equivalence relation. Conversely, let R be is 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 smallest 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).
One must not confuse equality and equation, although they are written similarly. An equality is an assertion, while an equation is the problem of finding values of some variables, called unknowns, to get an equality. Equation may also refer to an equality relation that is satisfied only for the values of the variables that one is interested on. For example x2 + y2 = 1 is the equation of the unit circle.
The etymology of the word is from the Latin aequalis, meaning uniform or identical, from aequus, meaning "level, even, or just."
Equality is always defined such that things that are equal have all and only the same properties. Some people define equality as congruence. Often equality is just defined as identity.
A stronger sense of equality is obtained if some form of Leibniz's law is added as an axiom; the assertion of this axiom rules out "bare particulars"—things that have all and only the same properties but are not equal to each other—which are possible in some logical formalisms. The axiom states that two things are equal if they have all and only the same properties. Formally:
In this law, the connective "if and only if" can be weakened to "if"; the modified law is equivalent to the original.
Instead of considering Leibniz's law as an axiom, it can also be taken as the definition of equality. The property of being an equivalence relation, as well as the properties given below, can then be proved: they become theorems. If a=b, then a can replace b and b can replace a.
Some basic logical properties of equality
The substitution property states:
- For any quantities a and b and any expression F(x), if a = b, then F(a) = F(b) (if either side makes sense, i.e. is well-formed).
Some specific examples of this are:
- For any real numbers a, b, and c, if a = b, then a + c = b + c (here F(x) is x + c);
- For any real numbers a, b, and c, if a = b, then a − c = b − c (here F(x) is x − c);
- For any real numbers a, b, and c, if a = b, then ac = bc (here F(x) is xc);
- For any real numbers a, b, and c, if a = b and c is not zero, then a/c = b/c (here F(x) is x/c).
The reflexive property states:
- For any quantity a, a = a.
This property is generally used in mathematical proofs as an intermediate step.
The symmetric property states:
The transitive property states:
The binary relation "is approximately equal" between real numbers or other things, even if more precisely defined, is not transitive (it may seem so at first sight, but many small differences can add up to something big). However, equality almost everywhere is transitive.
Although the symmetric and transitive properties are often seen as fundamental, they can be proved, if the substitution and reflexive properties are assumed instead.
Relation with equivalence and isomorphism
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.
Similarly, the sets
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
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, and is one motivation for the development of category theory.