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Formally, if M is a set, we define the identity function id_{M} on M to be that function with domain and codomain M which satisfies
If f : M → N is any function, then we have f o id_{M} = f = id_{N} o f. In particular, id_{M} is the identity element of the monoid of all functions from M to M.
When choosing M equal to the positive integers, one obtains the identity function Id(n), which is a multiplicative function considered in number theory.
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