ID # 1226043119 - Written Homework Week #2

ID # 1226043119 - Written Homework Week #2 1. Identify each of the following statements as true of false and explain your reasoning in full sentences. The domain of discourse is the integers. (a) ∃ x ∀ y ( y > x ) A. False. The statement says that there exists an x for which all y>x. However, this is not possible for any x as an integer like y=x or y=x-1 etc will be equal or smaller than x. (b) ∀ y ∃ x ( y > x ) A. True. The statement says that for all y there is an x such that y>x. This is true as for any y we can get an x = y-1 or x=y-(ANY +ve Integer), so that y>x. (c) ∃ x ( x = 0 → x = 1) A. False. The statement says that there exists an x where if x=0 then x=1. However, if the premise(x=0) is true, then x cannot be 1 and the conclusion is false. Thus the implication is false. 2. Write the formal negation of the following statements. Your negation must not contain any explicit negation symbols. (a) ∀ x ∃ y (2 < x ≤ y ) A. ∃x∀y((2≥x) ∨ (x>y)) (b) ∀ y ∃ x ( y > 0 → x ≤ 0) A. ∃y∀x((y>0) ∧ ( x>0)) 3. Negate verbally: (a) ”All people weigh at least 100 pounds.” A. Some people weigh at less than 100 pounds. (b) ”Somebody did not see any animals in the zoo” A. Everybody saw some animals in the zoo. 4. If P and Q are predicates over some domain, and if it is true that ∀ x ( P ( x ) ∨ Q ( x )), must ∀ xP ( x ) ∨ ∀ xQ ( x ) also be true? Explain. A. No, this is not the case. This is because if ∀ x(P (x) ∨ Q(x)) is true, this means that for all x, P(x) or Q(x) is true, whereas ∀ xP (x) ∨ ∀ xQ(x) means that for all x, P(x) is true or for all x, Q(x) is true. To explain, take ∀ x(P (x) ∨ Q(x)) as true. This implies that it could be the case that for some x = x 1 P(x) is true but Q(x) is not and for some x = x 2 Q(x) is true but P(x) is not. Now, the existence of x 1 implies that ∀ xQ ( x ) is false and x 2 implies that ∀ xP ( x ) is false. Thus, ∀ xP (x) ∨ ∀ xQ(x) is false. 5. Suppose P is the predicate defined by P ( x, y ) = x is friends with y , where x and y are people. (No one is considered to be friends with themselves.)