Logical Equivalences

Show that the following are true:

1. The conditional is logically equivalent to its contrapositive: $p\to q \equiv \neg q \to \neg p$.
2. The converse is logically equivalent to the inverse: $q\to p \equiv \neg p \to \neg q$.

Determine whether the following two statements are logically equivalent: $\neg(p \to q)$ and $p \wedge \neg q$. Explain how you know you are correct.

Are the statements $p \to (q\vee r)$ and $(p \to q) \vee (p \to r)$ logically equivalent?

Use a truth table to show that $(p \to q) \land (p \to r)$ and $p \to (q \land r)$ are logically equivalent.

Simplify the following statements (so that negation only appears right before variables).

1. $\neg(p \to \neg q)$.
2. $(\neg p \vee \neg q) \to \neg (\neg q \wedge r)$.
3. $\neg((p \to \neg q) \vee \neg (r \wedge \neg r))$.

Use DeMorgan's Laws, and any other logical equivalence facts you know to simplify the following statements. Show all your steps. Your final statements should have negations only appear directly next to the sentence variables or predicates ($p$, $q$, etc.), and no double negations. It would be a good idea to use only conjunctions, disjunctions, and negations.

1. $\neg((\neg p \wedge q) \vee \neg(r \vee \neg s))$.
2. $\neg((\neg p \to \neg q) \wedge (\neg q \to k))$ (careful with the implications).
3. $(p\land q) \to (p \lor q)$