Past exam of the year 1 of the computer science course of the University of Oxford / 2023 / Discrete mathematics, probability and continuous mathematics / 2 / d / v / Solution

Fix $b\in B$. Since $g\circ f$ is surjective, there is $a\in A$ with $g(f(a))=g(b)$. Injectivity of $g$ gives $f(a)=b$, proving that $f$ is surjective. All five conclusions are instances of injectivity and surjectivity under composition.