Answer the following items. Show your complete answer on a separate sheet of paper. Prove that the following sentences are tautologies. 1. p →p 2. p → (p V q) 3. [p Λ (p → q)] → q 4. p V ~p 5. q → (p V ~p) 6. ~p → (p →q) 7. (p Λ q) → p 8. (p → q) → [(p V r) → (q V r)] 9. ~q → ~(q Λ r)

$1. \, p \to p = \,\,\,\ \thicksim p \bigvee p = T\\ 2. \,p \to (p \bigvee q) = \,\,\,\ \thicksim p \bigvee p \bigvee q = T \bigvee q = T\\ 3.\, [p \bigwedge (p \to q)] \to q = \,\,\,\ \thicksim {[p \bigwedge (p \to q)]} \bigvee q = \\ \,\,\,\ \thicksim p \bigvee \thicksim (p \to q) \bigvee q =\\ \,\,\,\ \thicksim p \bigvee \thicksim (\thicksim p \bigvee q) \bigvee q = \\ \,\,\,\ \thicksim p \bigvee (p \bigwedge \thicksim q) \bigvee q = \\ \,\,\,\ (\thicksim p \bigvee p) \bigwedge (\thicksim p \bigvee \thicksim q) \bigvee q = \\ \,\,\,\ T \bigwedge (\thicksim p \bigvee \thicksim q) \bigvee q = (\thicksim p \bigvee \thicksim q) \bigvee q = \\ \,\,\,\ \thicksim p \bigvee \thicksim q \bigvee q = \thicksim p \bigvee T = T\\ 4.\,p \bigvee \thicksim p = \,\,\,\ \thicksim (\thicksim p \bigwedge p) = \,\,\,\ \thicksim F = T\\ 5.\, q \to (p \bigvee \thicksim p) = q \to T =\,\,\,\thicksim q \bigvee T = T\\ 6. \thicksim p \to (p \to q) = \,\,\, \thicksim p \to (\thicksim p \bigvee q) = \\ \,\,\,\ p \bigvee \thicksim p \bigvee q = \\ \,\,\,\ T \bigvee q = T\\ 7.\, (p \bigwedge q) \to p = \,\,\,\ \thicksim (p \bigwedge q) \bigvee p =\\ \,\,\,\ \thicksim p \bigvee \thicksim q \bigvee p = \\ \,\,\,\ T \bigvee \thicksim q = T\\ 8.\, (p \to q) \to [(p \bigvee r) \to (q \bigvee r)] = \\ \,\,\,\ (\thicksim p \bigvee q) \to [\thicksim (p \bigvee r) \bigvee (q \bigvee r)] = \\ \,\,\,\ (\thicksim p \bigvee q) \to [(\thicksim p \bigwedge \thicksim r) \bigvee (q \bigvee r)] = \\ \,\,\,\ \thicksim (\thicksim p \bigvee q) \bigvee [(\thicksim p \bigwedge \thicksim r) \bigvee (q \bigvee r)] = \,\,\,\ \\ \,\,\,\ (p \bigwedge \thicksim q) \bigvee (\thicksim p \bigwedge \thicksim r) \bigvee q \bigvee r = \\ \,\,\,\ (p \bigwedge \thicksim q) \bigvee q \bigvee (\thicksim p \bigwedge \thicksim r) \bigvee r\ = \\ \,\,\,\ [(p \bigvee q) \bigwedge (\thicksim q \bigvee q)] \bigvee [(\thicksim p \bigvee r) \bigwedge (\thicksim r \bigvee r)] = \\ \,\,\,\ [(p \bigvee q) \bigwedge T] \bigvee [(\thicksim p \bigvee r) \bigwedge T] = \\ \,\,\,\ p \bigvee q \bigvee \thicksim p \bigvee r = \\ \,\,\,\ T \bigvee q \bigvee r = T\\ 9.\, \thicksim q \to \,\,\, \thicksim (q \bigwedge r) = q \bigvee \thicksim q \bigvee \thicksim r = \\ \,\,\,\ T \bigvee \thicksim r = T\\$