Past exam of the year 1 of the computer science course of the University of Oxford / 2016 / Discrete mathematics, continuous mathematics and probability (resit) / 1 / c / Solution

A function $f:A\to B$ is onto, or a surjective function, when every $b\in B$ has a preimage: $\forall b\in B\exists a\in A:f(a)=b$. It is one-to-one, or an injective function, when $f(a)=f(a^{\prime })$ implies $a=a^{\prime }$ for all $a,a^{\prime }\in A$. It is bijective, a bijection, when both properties hold.

An inverse function is a function $f^{-1}:B\to A$ satisfyingFor a bijection, $f^{-1}(b)$ is the unique $a$ with $f(a)=b$: existence follows from surjectivity and uniqueness from injectivity. This is a two-sided inverse function, rather than just a left or right inverse.