Answer to Question #277825 in Math for Yangyang

Question

Use truth tables in Nos. 1 – 6.

1. Prove (p Λ p) <=> p. 4. Prove [(p Λ q) Λ r] <=> [p Λ (q Λ r)].

2. Prove (~p) <=> [~(~(~p))]. 5. If s<=> t, show that t <=>s.

3. Prove (p Λ q) <=> (q Λ p). 6. If s <=> t and t <=> u , show that s <=>u.

7. A young man received the following note from his girlfriend. "Dear Fred, I want to explain that I was really joking when I told you that I didn’t mean what I said about reconsidering my decision not to change my mind.” Did she finally change her mind or didn’t she? [Hint: Let P = “She changed her mind.” Apply Theorem 1 to [~(~(~(~(P))))].

8. In what context may the sentence, “Mr. Jones is the father of my best friend,” be considered to be equivalent to the sentence, “My best friend is a son of Mr. Jones.” Under what circumstances are these two sentences not equivalent?