(a) Each of the following true statements contains an implication. (i) For all a, b ∈ Z, if a − b is even then a + b is even. (ii) For all A ⊆ N, if A is finite then Ac is infinite. Write down the converse of each statement and decide (with justification) whether it is true or false

Use truth sets to prove or disprove the following quantified statements. a) Let D = {1, 3, 4} Let P(x) be the predicate "x2 + 2x > 0". Show that Va E D, x? + 2x > 0 b) Let D = {- 1, 0, 1, 3, 4} Let P(x) be the predicate "x + 2x > 0". Show that Vx E D, x? + 2x > 0 is false.

Let x e {1, 2} and y e {1, 4}, consider a predicate P(x, y) defined as P(x, y): 4 divides (x² + 3y). Rewrite the expression 3y[VxP(x, y)] by eliminating the quantifiers and replace P(x, y) with 4 divides (x² + 3y). Hence, determine the truth value of the statement.

Evaluate the proposition ∀∃y.P(x,y) ⇐⇒ ∃y.∀x.P(x,y) for an unknown predicate P. Could the statement be true? Must the statement be true for all P? Give a specific P (with domains for x and y) where it is either true or false. Prove that it is true for all P, or show a contradiction.

1. (a) Each of the following true statements contains an implication. (i) For all a, b E Z, if a – b is even then a + b is even. (ii) For all A C N, if A is finite then Aº is infinite. Write down the converse of each statement and decide (with justifica- tion) whether it is true or false. (b) Consider the statement: For all x, y, z Z. At least one of x – y, x – % and y – z is even. (i) Write down a roadmap for a proof of this statement by contradic- tion. (ii) Fill in the details of your roadmap to prove the statement.

If p is true, q is true, and r is false, determine whether each of the following compound statements aretrue or false”(a) p ∧ q(b) ∼ (q ∨ r)(c) r → (p ∧ q)(d) (p ∧ r) ∨ (p ∧ q → r)

(b) Consider the following statement: For all a € R, {x ≤ Q : x² = a} = 0 i. Which of "if", "only if", "iff" in your claim. ii. Which of "if", "only if", "iff" in your claim. a & Q. make this a true statement? Prove make this a false statement? Prove

(b) Consider the statement: For all x, y, z E Z. At least one of x - y, x – z and y – z is even. (i) Write down a roadmap for a proof of this statement by contradic- tion. (ii) Fill in the details of your roadmap to prove the statement.