Identity
From AoPSWiki
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, 
is an identity, since it holds regardless of the choice of variable. Therefore, it is sometimes written 
.
Abstract Algebra
Given a binary operation 
on a set 
, 
, an identity for is an element 
such that for all 
, 
. For example, in the real numbers, if we take to be the operation of multiplication (
), the number 
will be the identity for . If we instead took to be addition (
), 
would be the identity.
Identities in this sense are unique. Imagine we had two identities, 
and 
, for some operation . Then 
, so 
, and so and are in fact equal.
