Difference between revisions of "Identity"
ComplexZeta (talk | contribs) |
|||
Line 2: | Line 2: | ||
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. | ||
+ | |||
+ | == 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. | ||
Line 9: | Line 11: | ||
This usage of the word identity is common in [[abstract algebra]]. | 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." ( | + | 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 13:27, 12 July 2006
This article is a stub. Help us out by expanding it.
There are at least two possible meanings in mathematics for the word identity.
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
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, is an identity, since it holds regardless of choice of variable. We therefore sometimes write
.