Question 4. [32 MARKS]
Recall that we can prove that statements A and B are logically equivalent as follows:
1. Develop a proof of A ? B:
Suppose A
B
2. Develop of proof of B ? A:
Suppose B
A
For each of the following pairs of expressions, either prove they are equivalent by using the above technique, or
prove they are not equivalent by providing a counterexample world. (Hint: you can let the universe of discourse be
the set of natural numbers, and provide an example of properties P(x), Q(x) for which one expression is True and
the other is False.)
1. $(\forall x, P(x)) \implies (\forall x, Q(x))$ and $\forall x, P(x) \implies Q(x)$
2. $(\exists x, P(x)) \land (\exists x, Q(x))$ and $\exists x, P(x) \land Q(x)$
3. $(\forall x, P(x)) \lor (\forall x, Q(x))$ and $\forall x, P(x) \lor Q(x)$
4. $(\exists x, P(x)) \lor (\exists x, Q(x))$ and $\exists x, P(x) \lor Q(x)$
5. $(\forall x, P(x)) \implies (\forall x, Q(x))$ and $\forall x, P(x) \implies Q(x)$
6. $(\exists x \in A, P(x)) \land (\exists x \in B, P(x))$ and $\exists x \in A \cap B, P(x)$
7. $(\exists x \in A, P(x)) \lor (\exists x \in B, P(x))$ and $\exists x \in A \cup B, P(x)$
