Let A be a set of formulas such that: a) Every finite subset of A is satisfiable and b) For every formula a, either a EA or (a) E 4. The truth assignment v is defined: (T if A E A F if A & A for each symbol of the statement A. Prove that for every formula , v (p) = V if and only if p E 4. v(A): =

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.

When writing proofs by contradiction we begin by assuming the opposite of a statement, and then show that this leads to (i.e. entails) a contradiction. Which of the following statements constitutes a contradiction (that is, which of the following evaluates to false)? Suppose R(x, y) is a binary relation that relates two natural numbers. Select all that apply. A. O (3æ e N.R(x, æ) V R(x + 1, æ + 1)) ^ (Væ.¬R(x, æ)) B. O (Væ, y E N.R(x, y) → -R(y, x)) A R(3, 3) C. O (3r, y E N.R(x, y) → R(y, x)) ^ R(3, 4) A R(4, 3) D. O Va, y, z E N. (-(R(x,y) ^ R(y, z)) V R(x, z)) ) A R(2, 4) ^ R(4, 6) E. O None of the above

Parts a, h, and j

(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