Identity

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

Equations

An 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 the choice of variable. Therefore, it is sometimes written $(x+1)^2\equiv x^2+2x+1$ .

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 we instead 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.

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Identity

Equations

Abstract Algebra

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