:''For other uses, see identity (disambiguation).'' In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them. The term ''identity element'' is often shortened to ''identity'' when there is no possibility of confusion; we do so in this article. Let ''S'' be a set with a binary operation * on it. Then an element ''e'' of ''S'' is called a left identity if ''e'' * ''a'' = ''a'' for all ''a'' in ''S'', and a right identity if ''a'' * ''e'' = ''a'' for all ''a'' in ''S''. If ''e'' is both a left identity and a right identity, then it is called a two-sided identity, or simply an identity. For example, if (''S'',*) denotes the real numbers with addition, then 0 (number) is an identity. If (''S'',*) denotes the real numbers with multiplication, then 1 (number) is an identity. If (''S'',*) denotes the ''n''-by-''n'' square matrix (mathematics) with addition, then the zero matrix is an identity. If (''S'',*) denotes the ''n''-by-''n'' matrices with multiplication, then the identity matrix is an identity. If (''S'',*) denotes the set of all function (mathematics) from a set ''M'' to itself, with function composition as operation, then the identity map is an identity. If ''S'' has only two elements, ''e'' and ''f'', and the operation * is defined by ''e'' * ''e'' = ''f'' * ''e'' = ''e'' and ''f'' * ''f'' = ''e'' * ''f'' = ''f'', then both ''e'' and ''f'' are left identities, but there is no right or two-sided identity. As the last example shows, it is possible for (''S'',*) to have several left identities. In fact, every element can be a left identity. Similarly, there can be several right identities. But if there is both a right identity and a left identity, then they are equal and there is just a single two-sided identity. To see this, note that if ''l'' is a left identity and ''r'' is a right identity then ''l'' = ''l'' * ''r'' = ''r''. In particular, there can never be more than one two-sided identity. ==See also== *Inverse element *Additive inverse *Group (mathematics) *Monoid *Quasigroup Abstract algebra Algebra
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Words begining with Identity_element:
Identity_element
Identity element
:''For other uses, see identity (disambiguation).'' In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them. The term ''identity element'' is often shortened to ''identity'' when there is no possibility of confusion; we do so in this article. Let ''S'' be a set with a binary operation * on it. Then an element ''e'' of ''S'' is called a left identity if ''e'' * ''a'' = ''a'' for all ''a'' in ''S'', and a right identity if ''a'' * ''e'' = ''a'' for all ''a'' in ''S''. If ''e'' is both a left identity and a right identity, then it is called a two-sided identity, or simply an identity. For example, if (''S'',*) denotes the real numbers with addition, then 0 (number) is an identity. If (''S'',*) denotes the real numbers with multiplication, then 1 (number) is an identity. If (''S'',*) denotes the ''n''-by-''n'' square matrix (mathematics) with addition, then the zero matrix is an identity. If (''S'',*) denotes the ''n''-by-''n'' matrices with multiplication, then the identity matrix is an identity. If (''S'',*) denotes the set of all function (mathematics) from a set ''M'' to itself, with function composition as operation, then the identity map is an identity. If ''S'' has only two elements, ''e'' and ''f'', and the operation * is defined by ''e'' * ''e'' = ''f'' * ''e'' = ''e'' and ''f'' * ''f'' = ''e'' * ''f'' = ''f'', then both ''e'' and ''f'' are left identities, but there is no right or two-sided identity. As the last example shows, it is possible for (''S'',*) to have several left identities. In fact, every element can be a left identity. Similarly, there can be several right identities. But if there is both a right identity and a left identity, then they are equal and there is just a single two-sided identity. To see this, note that if ''l'' is a left identity and ''r'' is a right identity then ''l'' = ''l'' * ''r'' = ''r''. In particular, there can never be more than one two-sided identity. ==See also== *Inverse element *Additive inverse *Group (mathematics) *Monoid *Quasigroup Abstract algebra Algebra
See other meanings of words starting from letter:
I
IA | IB | IC | ID | IE | IF | IG | IH | IJ | IK | IL | IM | IN | IO | IP | IR | IS | IT | IU | IW | IX | IY | IZ |Words begining with Identity_element:
Identity_element
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