Assignment 2

# Assignment 2 - Assignment#2 Section 1.2 on page 19 5b)t e F...

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Assignment #2 Section 1.2 on page 19. 5b)t e) F f) T 7a) p => q, (~p) v q p | q | ~p | p => q | (~p) v q -------------------------------- T T F T T F T T T T T F F F F F F T T T 7b) P<=>Q and (P=>Q) ^ (Q=>P) p | q | p=>q | q=>p | p<=>q | (p=>q) ^ (q=>p) ---------------------------------------------- T | T | T | T | T | T F | T | T | F | F | F T | F | F | T | F | F F | F | T | T | T | T 7d) ~(P^Q) and (~P) V (~Q) P | Q | ~P | ~Q | P^Q | ~(P ^ Q) | (~P) v (~Q) ------------------------------------------------ T | T | F | F | T | F | F F | T | T | F | T | F | F T | F | F | T | T | F | F F | F | T | T | F | T | T 8g) B is invertible is a necessary and sufficient condition for det(B) <> 0. Logical connectives: (B is invertible) <=> (det(B) <> 0) 12a) converse: (f'(x) = 0) => ((f has a rel. min at x) ^ ( f is diff. at x)) False contrapositive: (f'(x) <>0) => ((f does have a rel. min at x) V ( f is not diff. at x)) True. 12B) Converse: If n=2 or n is odd, then n is prime. False. 9 is odd and not prime. Contrapositive: If n is even and not equal to 2, then n is not prime. True 12c) converse: ((x is real) ^ ( x is not rational)) => ( x is irrational) True.

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contrapositive: ((x is not real) V (x is rational)) => ( x is not irrational) True.12d) converse: (|x| = 1) => ((x = 1) v (x = -1)) True.contrapositive:(|x| neq 1) => ((x neq 1) ^ (x neq -1)) True. 14a) P | Q | P => Q | ~P | ~Q | ~P => ~Q ------------------------------------ T T T F F T T F F F T T F T T T F F F F T T T T 14b) P=>Q and its inverse are both true if both P and Q are either both true or false as shown in the truth table above 14c) Conditional Sentence : P => Q Converse : Q => P Inverse : ~P => ~Q Contrapositive : ~Q => ~P Contrapositive of inverse : ~(~Q) => ~(~P) or Q => P Inverse of contrapositive : ~(~Q) => ~(~P) or Q => P Hence, both the inverse of the contrapositive and the contrapositive of the inverse are equivalent to the converse. 13c)
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Assignment 2 - Assignment#2 Section 1.2 on page 19 5b)t e F...

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