Prove that the following compound propositions are equivalent:

  1. \neg (p \wedge q) \equiv \neg p \vee \neg q
  2. \neg (p \vee q) \equiv \neg p \wedge \neg q
  3. p \to q \equiv \neg p \vee q
  4. p \to q \equiv \neg q \to \neg p
  5. p \vee q \equiv \neg p \to q
  6. p \wedge q \equiv \neg (p \to \neg q)
  7. \neg(p \to q) \equiv p \wedge \neg q
  8. (p \to q) \wedge (p \to r) \equiv p \to (q \wedge r)
  9. (p \to r) \wedge (q \to r) \equiv (p \vee q) \to r
  10. (p \to q) \wedge (p \to r) \equiv p \to (q \vee r)
  11. (p \to r) \wedge (q \to r) \equiv (p \wedge) \to r
  12. (p \vee q) \vee r \equiv r \equiv p \vee (q \vee r) and (p \wedge q) \wedge r \equiv p \wedge (q \wedge r)
  13. p \vee (q \wedge r) \equiv (p \vee q) \wedge (p \vee r) and p \wedge (q \vee r) \equiv (p \wedge q) \vee (p \wedge r)
  14. p \vee (p \wedge q) \equiv p and p \wedge (p \vee q) \equiv p
  15. p \wedge q \equiv q \wedge p and p \vee q \equiv q \vee p
  16. T \wedge p \equiv p and F \vee p \equiv p
  17. T \vee p \equiv T and T \wedge p \equiv p