a) Consider the following proof of logical equivalence: Vx(P(x) Q(x)) = x(P(x) VQ(x)) =Vx(P(x) AQ(x)) =...

   

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a) Consider the following proof of logical equivalence: Vx(P(x) Q(x)) = x(P(x) VQ(x)) =Vx(P(x) AQ(x)) = x(P(x) AQ(x))) =(3xP(x) xQ(x)) =(3xP(x)) V (x(Q(x))) =(3xP(x)) V (VxQ(x)) =(3xP(x)) > (VxQ(x)) Is the proof correct or not? If not, point out where the mistake is. (10 Points) Prove the following statements: b) If an integer t is divisible by 3, then 2t+1 is not divisible by 3. (5 Points) c) Use induction to show that k=1 holds for all integer n > 1. (5 Points) 2 3 2n+3 = k(k+2) 2 (n+1)(n+2) (3) (4) (5) (6) OE a) Consider the following proof of logical equivalence: Vx(P(x) Q(x)) = x(P(x) VQ(x)) =Vx(P(x) AQ(x)) = x(P(x) AQ(x))) =(3xP(x) xQ(x)) =(3xP(x)) V (x(Q(x))) =(3xP(x)) V (VxQ(x)) =(3xP(x)) > (VxQ(x)) Is the proof correct or not? If not, point out where the mistake is. (10 Points) Prove the following statements: b) If an integer t is divisible by 3, then 2t+1 is not divisible by 3. (5 Points) c) Use induction to show that k=1 holds for all integer n > 1. (5 Points) 2 3 2n+3 = k(k+2) 2 (n+1)(n+2) (3) (4) (5) (6) OE