3. Use logical equivalences to demonstrate that the inverse and the converse
of p -> q are logically equivalent. Identify all logical equivalences by name.
You will not receive credit for a truth table solution.
4. Rephrase verbally in equivalent only if, sufficient, necessary, contrapositive
and unless form: "if we had an FTL drive, then we could visit the stars".
5. Is the statement ∃x∀y(xy = 0) true or false? Explain.
6. If P and Q are predicates over some domain, and if it is true that ∀x(P(x) ∨
Q(x)), must ∀xP(x) ∨ ∀Q(x) also be true? Explain.
7. Suppose P is the predicate defined by P(x, y) = x is friends with y, where
x and y are people. (No one is considered to be friends with themselves.)
Translate the formal expression ∀x∃y∃z(y ≠ z ∧ P(x, y) ∧ P(x, z)) into
English.
8. Let P be defined as in the previous problem. Is ∀x∃y∃z(y ≠ z → P(x, y) ∧
P(x, z)) true or false? Explain.