Equality (mathematics)

In mathematics, two mathematical objects are considered equal if they are precisely the same in every way. This defines a binary predicate, equality, denoted "="; x = y iff x and y are equal. Equivalence in the general sense is provided by the construction of a equivalence relation between two elements. A statement that two expressions denote equal quantities is an equation.

Beware that sometimes a statement of the form "A = B" may not be an equality. For example, the statement T(n) = O(n²) means that T(n) grows at the order of n². It is not an equality, because the sign "=" in the statement is not the equality sign; indeed, it is meaningless to write O(n²) = T(n). See Big O notation for more on this.

Given a set A, the restriction of equality to the set A is a binary relation, which is at once reflexive, symmetric, antisymmetric, and transitive. Indeed it is the only relation on A with all these properties. Dropping the requirement of antisymmetry yields the notion of equivalence relation. Conversely, given any equivalence relation R, we can form the quotient set A/R, and the equivalence relation will 'descend' to equality in A/R. Note that it may be impractical to compute with equivalence classes: one solution often used is to look for a distinguished normal form representative of a class.

Predicate logic contains standard axioms for equality that formalise Leibniz's law, put forward by the philosopher Gottfried Leibniz in the 1600s. Leibniz's idea was that two things are identical if and only if they have precisely the same properties. To formalise this, we wish to say

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

However, in first order logic, we cannot quantify over predicates. Thus, we need to use an axiom schema:

Given any x and y, if x equals y, then P(x) iff P(y).

This axiom schema, valid for any predicate P in one variable, takes care of only one direction of Leibniz's law; if x and y are equal, then they have the same properties. We can take care of the other direction by simply postulating:

Given any x, x equals x.

Then if x and y have the same properties, then in particular they are the same with respect to the predicate P given by P(z) iff x = z. Since P(x) holds, P(y) must also hold, so x = y.

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).

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 from the substitution and reflexive properties.