Difference between revisions of "Identity"
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There are at least two possible meanings in mathematics for the word '''identity'''. | There are at least two possible meanings in mathematics for the word '''identity'''. | ||
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== Equations == | == Equations == | ||
− | + | An '''identity''' is a general relationship which always holds, usually over some choice of [[variable]]s. For example, <math>(x+1)^2=x^2+2x+1</math> is an identity, since it holds regardless of the choice of variable. Therefore, it is sometimes written <math>(x+1)^2\equiv x^2+2x+1</math>. | |
− | An '''identity''' is | ||
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== Abstract Algebra == | == Abstract Algebra == | ||
+ | Given a [[binary operation]] <math>G</math> on a [[set]] <math>S</math>, <math>G: S\times S\to S</math>, an identity for <math>G</math> is an [[element]] <math>e\in S</math> such that for all <math>a\in S</math>, <math>G(e,a)=G(a,e)=a</math>. For example, in the [[real number]]s, if we take <math>G</math> to be the [[operation]] of [[multiplication]] (<math>G(a,b)=a\cdot b</math>), the number <math>1</math> will be the identity for <math>G</math>. If we instead took <math>G</math> to be addition (<math>G(a, b) = a + b</math>), <math>0</math> would be the identity. | ||
− | + | Identities in this sense are [[unique]]. Imagine we had two identities, <math>e</math> and <math>e'</math>, for some operation <math>G</math>. Then <math>e=G(e,e')=e'</math>, so <math>e=e'</math>, and so <math>e</math> and <math>e'</math> are in fact equal. | |
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− | Identities in this sense are unique. | ||
− | + | ==See Also== | |
+ | *[[Operator inverse]] | ||
− | + | [[Category:Algebra]] | |
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− | [[Category: | ||
[[Category:Abstract algebra]] | [[Category:Abstract algebra]] | ||
− | [[Category: | + | [[Category:Definition]] |
Latest revision as of 13:38, 14 July 2021
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.