Problem 3. (12 points)
(a) Distribute the quantifiers for the statement 3x (P(x)V -.Q(x)) so that all the quantifiers occur immediately before the predicates or the negation symbol. If distribution is not possible, briefly explain why.
(b) Determine whether the two statements 3x(P(x) —> Q(x)) and 3xP(x) 3rQ(x) are equivalent. Explain.
Problem 4. (10 points) Express the negation of 3xVy(P(x, y) V Q(y)) so that the negation symbols immediately precede the predicates. Show the work of the derivation and state what rules are used.
Problem 5. (12 points) Assume the following statements are true for some propositions p, q, and r: V -,q) r) Can we definitively say that p is true? Why or why not?
Problem 6. (15 points) Let n be a real number. Consider the following statement: “If n > 1, then TO > 1.”
(a) Rewrite the statement as a proposition using predicate variables p and q. Be sure to specify what p and q mean in your proposition.
(b) Determine whether the following arguments are valid assuming the statement is true. Explain why or why not (i) “If 712 < 1, then n < 1. (ii) “Suppose that n2 > 1. Then n > 1.”
Problem 7. (15 points) Prove or disprove that the equation 3×2 + 2y2 = 22 has no positive integer solutions.
Problem 8. (18 points) In this question, we will prove that 121x + 23 is even iff (if and only if) x is odd.
(a) Prove if x is odd, then 121x + 23 is even using a direct proof.
(b) Prove if 121x + 23 is even, then x is odd using any proof technique. State the proof technique you used. Additional note: In part
(a), you showed the t— direction. In part
(b), you showed the —> direction. When proving something in the form p to q, it is very common to take this two step approach.