Identity

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There are at least two possible meanings in mathematics for the word identity.

Abstract Algebra

Given a binary operation G on a set S, $G: S \times S \to S$ , an identity for G is an element $e\in S$ such that for all $a \in S$ , $G(e, a) = G(a, e) = a$ . For example, in the real numbers, if we take G to be the operation of multiplication $G(a, b) = a\cdot b$ , the number 1 will be the identity for G. If instead we took G to be addition ( $G(a, b) = a + b$ ), 0 would be the identity.

Identities in this sense are unique. Imagine we had two identities, $e$ and $e'$ , for some operation $G$ . Then $e = G(e, e') = e'$ , so $e = e'$ , and so e and e' are in fact equal.

This usage of the word identity is common in abstract algebra.

Equations

An alternative meaning for the word identity is "a general relationship which always holds, usually over some choice of variables." For example, $(x+1)^2=x^2+2x+1$ is an identity, since it holds regardless of choice of variable. We therefore sometimes write $(x+1)^2\equiv x^2+2x+1$ .

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Identity

Abstract Algebra

Equations