Chapter 1 - Section 1.4 - Predicates and Quantifiers - Exercises - Page 53: 9

a) $\exists x(P(x)\land Q(x))$ b) $\exists x(P(x)\land \neg Q(x))$ c) $\forall x(P(x)\lor Q(x))$ d) $\neg \exists x(P(x)\lor Q(x))$

We replace the quantifiers first, with "there is" going to $\exists$, "every" going to $\forall$, and "no" going to $\neg \exists$. Then, we replace the statements with $P(x)$ and $Q(x)$ to obtain: a) $\exists x(P(x)\land Q(x))$ b) $\exists x(P(x)\land \neg Q(x))$ c) $\forall x(P(x)\lor Q(x))$ d) $\neg \exists x(P(x)\lor Q(x))$