Identity element: Difference between revisions
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In [[algebra]], an '''identity element''' or '''neutral element''' with respect to a [[binary operation]] is an element which leaves the other operand unchanged, generalising the concept of [[zero]] with respect to [[addition]] or [[one]] with respect to [[multiplication]]. | In [[algebra]], an '''identity element''' or '''neutral element''' with respect to a [[binary operation]] is an element which leaves the other operand unchanged, generalising the concept of [[zero]] with respect to [[addition]] or [[one]] with respect to [[multiplication]]. | ||
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==Examples== | ==Examples== | ||
* Existence of an identity element is one of the properties of a [[group (mathematics)|group]] or [[monoid]]. | * Existence of an identity element is one of the properties of a [[group (mathematics)|group]] or [[monoid]]. | ||
* An [[identity matrix]] is the identity element for [[matrix multiplication]]. | * An [[identity matrix]] is the identity element for [[matrix multiplication]]; a [[zero matrix]] is the identity element for [[matrix addition]]. | ||
* The [[empty set]] is the identity element for set [[union]]. | |||
==See also== | ==See also== | ||
* [[Identity (mathematics)]] | * [[Identity (mathematics)]] | ||
Latest revision as of 15:33, 8 December 2008
In algebra, an identity element or neutral element with respect to a binary operation is an element which leaves the other operand unchanged, generalising the concept of zero with respect to addition or one with respect to multiplication.
Formally, let be a binary operation on a set X. An element I of X is an identity for if
holds for all x in X. An identity element, if it exists, is unique.
Examples
- Existence of an identity element is one of the properties of a group or monoid.
- An identity matrix is the identity element for matrix multiplication; a zero matrix is the identity element for matrix addition.
- The empty set is the identity element for set union.