Past exam of the year 1 of the computer science course of the University of Oxford / 2019 / Discrete mathematics, continuous mathematics, and probability resit / 1 / d / Solution

Put $B=T\cap S$. By distributivity, $g(A)=B\cup (T\cap A)$ and $B\subseteq T$. Consequently,Here we used associativity, distributivity, and idempotence of set intersection and set union. The equality holds for every $A\in \mathcal{P}(U)$, so $g\circ g=g$: $g$ is an idempotent function.