1. For each of the following, first write the argument in symbolic form, and then determine if it is valid or invalid. If the argument is valid, prove it using known logical equivalences and inference rules. If it is invalid, demonstrate that by giving a counter example.

(a) If I do not start planning early, or go to Hawaii in February, then our vacation is not affordable and we take a driving trip. If we take a driving trip, then we eat at Tim Horton's. We did not eat at Tim Horton's. Therefore, we did not go to Hawaii in February.

(b) I go hiking or play golf. If I go hiking, then if I'm tired then we go to the pub for dinner. If I don't get enough sleep, then I'm tired. Therefore, if we go to the pub for dinner then I don't get enough sleep.

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2. Suppose the universe of all variables is U = {1,2, 3}. (a) Write a statement involving quantifiers which is logically equivalent to [(3 ·1 = 2.1)] A (3 ·1 = 2- 2) A (3 ·1 = 2 . 3)] V[(3· 2 = 2.1)] A (3- 2 = 2- 2) A (3 -2 = 2 - 3)] V[(3 · 3 = 2.1)] A (3 -3 = 2. 2) A (3 - 3 = 2 - 3)] %3D (b) Write a statement not involving quantifiers which is logically equivalent to. Vx, 3y, 3x + 2y = 5. (c) What is the truth value of the statement Va, 3y, (æ + 1) → (x < y)? Explain. (d) What is the truth value of the statement 3x, Vy, (x 1) (x < y)? Explain. 3. (a) Let p(x) be "x has passed Math 122." Suppose the universe of x is the set of all students. Write Vr, (x Gary) p(x)) A -p(Gary) in plain English. Do not literally translate the symbols into words. (b) Suppose p(n) and q(n) are statements involving the integer n. Explain why 3n, p(n)^q(n) is not logically equivalent to [3n, p(n)] ^ [3n, q(n)]. (Hint: check it out with p(n): "n is even" and q(n): "n is odd.") (c) Suppose a(n) and b(n) are statements involving the integer n. Is Vn, a(n) Ab(n) logically equivalent to [Vn, a(n)] ^ [Vn, b(n)]. Why or why not? (d) Let p(n) be an open statement. Suppose the universe of n is empty (i.e., contains no elements). What is the truth value of Vn, p(n)? What about 3n, p(n)? Explain. 2. Suppose the universe of all variables is U = {1,2, 3}. (a) Write a statement involving quantifiers which is logically equivalent to [(3 ·1 = 2.1)] A (3 ·1 = 2- 2) A (3 ·1 = 2 . 3)] V[(3· 2 = 2.1)] A (3- 2 = 2- 2) A (3 -2 = 2 - 3)] V[(3 · 3 = 2.1)] A (3 -3 = 2. 2) A (3 - 3 = 2 - 3)] %3D (b) Write a statement not involving quantifiers which is logically equivalent to. Vx, 3y, 3x + 2y = 5. (c) What is the truth value of the statement Va, 3y, (æ + 1) → (x < y)? Explain. (d) What is the truth value of the statement 3x, Vy, (x 1) (x < y)? Explain. 3. (a) Let p(x) be "x has passed Math 122." Suppose the universe of x is the set of all students. Write Vr, (x Gary) p(x)) A -p(Gary) in plain English. Do not literally translate the symbols into words. (b) Suppose p(n) and q(n) are statements involving the integer n. Explain why 3n, p(n)^q(n) is not logically equivalent to [3n, p(n)] ^ [3n, q(n)]. (Hint: check it out with p(n): "n is even" and q(n): "n is odd.") (c) Suppose a(n) and b(n) are statements involving the integer n. Is Vn, a(n) Ab(n) logically equivalent to [Vn, a(n)] ^ [Vn, b(n)]. Why or why not? (d) Let p(n) be an open statement. Suppose the universe of n is empty (i.e., contains no elements). What is the truth value of Vn, p(n)? What about 3n, p(n)? Explain.