Past exam of the year 1 of the computer science course of the University of Oxford / 2022 / Digital systems, linear algebra, introduction to formal proof resit / 7 / d / Solution

Simultaneous propositional substitution givesLet $K=\neg (X\wedge Y)$. The first disjunct equals $K\vee \neg (X\vee Y)$. By De Morgan's laws, $\neg (X\vee Y)=\neg X\wedge \neg Y$ implies $K=\neg X\vee \neg Y$, so absorption reduces that first disjunct to $K$. The second disjunct is $\neg Y\wedge K$ and is also absorbed by $K$. ThereforeThe resulting Boolean function is false only when both $X$ and $Y$ are true, providing a four-row truth table check.