Identity operator

In mathematics, an identity function, also called an identity relation or identity map or identity transformation, is a function that always returns the same value that was used as its argument. In equations, the function is given by $f (x) = x$ .

Formally, if $M$ is a set, the identity function $f$ on $M$ is defined to be that function with domain and codomain $M$ which satisfies

In other words, the function value $f (x)$ in $M$ (that is, the codomain) is always the same input element $x$ of $M$ (now considered as the domain). The identity function on $M$ is clearly an injective function as well as a surjective function, so it is also bijective.

The identity function $f$ on $M$ is often denoted by $id M$ .

In set theory, where a function is defined as a particular kind of binary relation, the identity function is given by the identity relation, or diagonal of $M$ .

If $f : M \to N$ is any function, then we have $f \circ id M = f = id N \circ f$ (where "∘" denotes function composition). In particular, $id M$ is the identity element of the monoid of all functions from $M$ to $M$ .

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