Equality (mathematics)

Loosely, equality is the state of being quantitatively the same. More formally, equality (or the identity relation) is the binary relation on a set X defined by

$\{(x, x)\;|\;x\in X\}$.

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 relation of equality is also antisymmetric. These four properties uniquely determine the equality relation on any set S and render equality the only relation on S that is both an equivalence relation and a partial order. It follows from this that equality is the smallest equivalence relation on any set S, in the sense that it is a subset of any other equivalence relation on S. An equation is simply an assertion that two expressions are related by equality (are equal).

In weakly typed programming languages like C, a logical operation of equality test often yields a value value of 1 or 0 or is automatically converted in such a value if the environment requires this. The mathematical equivalent of such operation is Kronecker delta. Languages with stronger type systems (like Java) often have a dedicated Boolean data type.

The etymology of the word is from the Latin aequalis, meaning uniform or identical, from aequus, meaning "level, even, or just."

Logical formulations

The equality relation 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:

Given any x and y, x = y if, given any predicate P, P(x) if and only if P(y).

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).

In first-order logic, this is a schema, since we can't quantify over expressions like F (which would be a functional predicate).

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:

• For any quantities a and b, if a = b, then b = a.

The transitive property states:

• For any quantities a, b, and c, if a = b and b = c, then a = c.

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

See also: Equivalence relation and Isomorphism

In some contexts, equality is sharply distinguished from equivalence or isomorphism.^[1] For example, one may distinguish 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

$\{\text{A}, \text{B}, \text{C}\} \,$ and $\{ 1, 2, 3 \} \,$

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

$\text{A} \mapsto 1, \text{B} \mapsto 2, \text{C} \mapsto 3.$

However, there are other choices of isomorphism, such as

$\text{A} \mapsto 3, \text{B} \mapsto 2, \text{C} \mapsto 1,$

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, and is one motivation for the development of category theory.

See also

• Equals sign
• Inequality
• Logical equality
• Extensionality

References

1. ^ (Mazur 2007)

• Mazur, Barry (12 June 2007), When is one thing equal to some other thing?, http://www.math.harvard.edu/~mazur/preprints/when_is_one.pdf 
• Mac Lane, Saunders; Garrett Birkhoff (1967). Algebra. American Mathematical Society.