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Difference between revisions of "Identity"

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There are at least two possible meanings in mathematics for the word '''identity'''.
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There are at least two possible meanings in mathematics for the word identity.
 
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== Equations ==
 
== Equations ==
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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>.
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An '''identity''' is "a general relationship which always holds, usually over some choice of variables."  For example, <math>(x+1)^2=x^2+2x+1</math> is an identity, since it holds regardless of choice of variable. We therefore sometimes write <math>(x+1)^2\equiv x^2+2x+1</math>.
 
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== Abstract Algebra ==
 
== Abstract Algebra ==
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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.
  
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Given a [[binary operation]] G on a [[set]] S, <math>G: S \times S \to S</math>, an identity for G 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 G to be the operation of [[multiplication]] <math>G(a, b) = a\cdot b</math>, the number 1 will be the identity for G.  If instead we took G to be addition (<math>G(a, b) = a + b</math>), 0 would be the identity.
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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. 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 e and e' are in fact equal.
 
  
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This usage of the word identity is common in [[abstract algebra]].
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==See Also==
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*[[Operator inverse]]
  
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See also the article [[Inverse with respect to an operation]].
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[[Category:Algebra]]
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[[Category:Abstract algebra]]
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[[Category:Definition]]

Latest revision as of 12:38, 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, $(x+1)^2=x^2+2x+1$ 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.

See Also

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