Determine when the given proposition is true

Question

The question asks to determine when the following proposition is true using truth-tables:

When I construct a truth-table, I have the following:

Thus for the whole proposition:

So, am I correct in saying that this proposition is never true?

Answer 1

It is correct. You could simplify the check by using some “propositional identities”: for instance, you can prove that $$ A\land(B\lor C)\equiv (A\land B)\lor(A\land C) $$ so in your case you can substitute $\lnot Q\land(\lnot P\lor Q)$ with $$ (\lnot Q\land\lnot P)\lor(\lnot Q\land Q) $$ Since $\lnot Q\land Q$ is always false, you can substitute the whole with $\lnot Q\land \lnot P$. Thus the first block becomes (equivalent to) $$ P\land(\lnot Q\land\lnot P) $$ Oh, well, this is false for every $P$ and $Q$, because of $$ X\land Y\equiv Y\land X \qquad\text{and}\qquad A\land(B\land C)\equiv (A\land B)\land C $$ so $$ P\land(\lnot Q\land\lnot P)\equiv P\land(\lnot P\land\lnot Q)\equiv (P\land\lnot P)\land\lnot Q $$