1. Use a truth tabte to determine whether (p → q) → r and p ∧ K (q ∨ ~ r) are equivalent.
2. Use a truth tabte to determine whether the argument form is valid.
~ p
q
∴ p → q
3. Sharky, a leader of the underworld, was killed by one of his own band of four henchmen. Detective Sharp interviewed the men and determined that all were lying except for one. He deduced who killed Sharky on the basis of the following statements:
(a) Socko: Lefty killed Sharky.
(b) Fats: Muscles didn‘t kill Sharky.
(c) Lefty: Muscles was shooting craps with Socko when Sharky was knocked off.
(d) Muscles: Lefty didn‘t kill Sharky.
Who did kill Sharky?
4. Given any integers a, b, and s, if a - b is odd and b - s is even, what can you say about the parity of a - s? Prove your answer.
5. Prove that for all real numbers r, if r2 is irrational, then r is irrational.