Question

6. Show that ~ (p → q) and p ∧~q are logically equivalent. (Hint: you can use a truth table to prove it or you apply De Morgan law to show the ~(p → q) is p ∧~q.

7.Let p and q be the propositions.
p: I bought a lottery ticket this week.
q: I won the million-dollar jackpot on Friday.
a) Form a tautology using p. Express the tautology in English sentence.
b) Form a tautology using q. Express the tautology in English sentence.
c) Form a contradiction using p. Express the contradiction in English sentence.
d) Form a contradiction using q. Express the contradiction in English sentence.

8. If you have a tautology r and you negate r, what kind of sentence do you get?
a. A tautology
b. A contradiction
c. A sentence that is neither a contradiction nor a tautology
d. You can’t tell—it could be any of (a), (b), or (c).