Give sequent proofs of each of the following sequents: 1. P^(Q⇒R), Q^(P→S) ⇒ R^S I. 2. PA~Q~(PQ) 3. (PAQ) V (R→S),(P→S)^R⇒S 4. ⇒(P→(Q→R))→ (Q→(P→R))

Determine whether (p→q)^r is equivalent to p(q^r). To get full credit on this question, you must show your work AND give an answer stating that the tables are equivalent or not equivalent. Consider a "1" to be true and a "0" to be false. p✈q p➜(q^r) Р 1 1 1 1 0 0 0 0 9 1 1 0 0 1 1 0 0 Are the two statements equivalent? r 1 0 1 0 1 0 1 0 (p→q)^r (q^r)

Prove this proposition algebraically

Is the following compound proposition satisfiable? Justify your answer. (p^q) (mvq) ^ ( q→ ¯p)

Prove ¬¬A := ¬(¬A) = A ¬(A ^ B) = (¬A) V (¬B) ¬(A V B) = (¬A) ^ (¬B) (Vx € X: E(x)) = (x € X : ¬E(x))

1. Consider the formula (P^Q) → (P^ (P→Q)) (a) Prove the validity of the above formula using rules of natural deduc- tion (b) Prove the validity of the above formula using semantic tableaux