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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.
 
There are at least two possible meanings in mathematics for the word identity.
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== Abstract Algebra ==
  
 
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.
 
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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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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== Equations ==
  
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An alternative meaning for the word identity is "a general relationship which always holds, usually over some choice of variables."  (This could undoubtedly be better-phrased.)
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An alternative meaning for the word 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>.

Revision as of 12:27, 12 July 2006

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

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 instead we 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 $e = e'$ so e and e' are in fact equal.

This usage of the word identity is common in abstract algebra.

Equations

An alternative meaning for the word 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 choice of variable. We therefore sometimes write $(x+1)^2\equiv x^2+2x+1$.

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