 Solution

Splitting the Boolean function according to the first tested Boolean variable givesThe identity $y\vee (\neg y\wedge z)=y\vee z$ follows by considering $y=0$ and $y=1$. Using it and distributing Boolean conjunction over Boolean disjunction,The first two terms combine to $y$ because $x\vee \neg x=1$. This also explains the truth table: $y=1$ always makes the output one, while with $y=0$ the output is one exactly when both $x$ and $z$ are one.