Difference between revisions of "Identity"
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This usage of the word identity is common in [[abstract algebra]]. | This usage of the word identity is common in [[abstract algebra]]. | ||
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[[Category:Elementary algebra]] | [[Category:Elementary algebra]] | ||
[[Category:Abstract algebra]] | [[Category:Abstract algebra]] | ||
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Revision as of 10:41, 23 November 2007
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 choice of variable. We therefore sometimes write 
.
Abstract Algebra
Given a binary operation G on a set S, 
, an identity for G is an element 
such that for all 
, 
. For example, in the real numbers, if we take G to be the operation of multiplication 
, the number 1 will be the identity for G. If instead we took G to be addition (
), 0 would be the identity.
Identities in this sense are unique. Imagine we had two identities, 
and 
, for some operation 
. Then 
, so 
, and so e and e' are in fact equal.
This usage of the word identity is common in abstract algebra.
