(a) Given a conditional statement r → p, find the inverse of its converse, and the inverse of it contrapositive. (b) Show that the conditional statements [(p V g) ^ (p → r) ^ (q→ r)] → r is a tautology by using truth tables.

For the following sets A C WHILE-data, state whether they are WHILE-decidable and explain your answer. In cases where A is decidable this explanation should consist of a description of the decision procedure. Recall that p denotes the encoding of WHILE-program p in D. WHILE i. A= {p| [p]" (d) = c} where c, d = D are fixed. ii. A = {˜ [p, k]¯ | p with input nil terminates in / steps or less }.

Prove the following proposition to be a tautology by constructing a proof for the following theorem within the system of sentential logic: ~(P ≡ Q) ⊃ (P ≡ ~Q)

a) Show that: (Vx)(A → (B = C)) → ((\x)(A → B) = (Vx)(A → → C)) b) Show that: (\x)((A v B) → C) → (Vx) (A → C) c) [Show that: (\x)(A → (B ^ C)) → (\x)(A → B) d) Prove (AV (Vx) BC = (\x)B(AV C), as long as x not free in A. (Hint: Translate first to Standard notation)

Use rules of inference to show that if Vx(P(x) v are true, then \x(¬R(x) → P(x)) Question 2. Q(x)) and Vx ((¬P(x) ^ Q(x)) → R(x))