bijection

Let $X$ and $Y$ be sets. A function $f\colon X\to Y$ that is one-to-one and onto is called a bijection or bijective function from $X$ to $Y$ .

When $X=Y$ , $f$ is also called a permutation of $X$ .

An important consequence of the bijectivity of a function $f$ is the existence of an inverse function $f^{-1}$ . Specifically, a function is invertible if and only if it is bijective. Thus if $f:X\rightarrow Y$ is a bijection, then for any $A\subset X$ and $B\subset Y$ we have

It easy to see the inverse of a bijection is a bijection, and that a composition of bijections is again bijective.