Identity function

Definition

Let $A$ be a set. The identity function of $A$ is the function $\mathrm{id}_A \colon A \to A$ with $\mathrm{id}_A(a) = a$ for each $a \in A$.

Remarks

For each function $f \colon A \to B$, $f \circ \mathrm{id}_A = f$ and $\mathrm{id}_B \circ f = f$. See function composition. The identity function is a bijection, and it is its own inverse. Its graph is the diagonal $\{(a, a) \mid a \in A\}$.

Explanations

L1 human

The identity function does nothing: each input comes out unchanged. It plays the role for composition that the number $1$ plays for multiplication.