1. The propositional variables, p, q, and s have the following truth assignments: p = T, q = T, s = F. Give the truth value for the following compound propositions: a. p q b. (p q) s c. p (q s) d. p (q s) e. (q p s) 2. Define the following propositions: p: I will return to college. q: I will not get a job. Translate the following English sentence into logical expressions using the definitions above: a. Not getting a job is a sufficient condition for me to return to college. b. If I return to college then I won't get a job c. I am not getting a job, but I am still not returning to college. d. I will return to college only if I won't get a job e. There's no way I am returning to college. f. I will get a job and return to college. 3. Predicates P and Q are defined below. The domain of discourse is the set of all positive integers. P(x): x is prime Q(x): x is a perfect square Are the following logical expressions propositions? If the answer is yes, indicate whether the statement is true or false. a. P(x) b. P(2) c. x (Q(x) P(x)) 4. Find a counter-example for each of the statements below: a. Every month of the year has 30 or 31 days. b. If n is an integer and n2 is divisible by 4, then n is divisible by 4. c. For every positive integer x, x3 < 2x d. Every positive integer can be expressed as the sum of the squares of two integers. e. The multiplicative inverse of a real number x, is a real number y such that xy = 1. Every real number has a multiplicative inverse. 5. Prove that if n is an odd integer, then n2 is an odd integer. State what method of proof you are using. At the beginning of each proof, state what you are assuming and what the proof will show. 6. Indicate whether the following statements are true or false: a. R 2 R 3 b. Z 2 R 2 c. If A B, then A2 B2 d. Z 3 Z 2 = 7. Define the following sets: A = {1, 2, 6} B = {2, 3, 4} C = {5} D = {x Z: 1 x 6} Do A, B, and C form a partition of D? If not, which condition of a partition is not satisfied? 8. Which of the following expressions are functions from R to R ? If f is a function, give its range. a. f ( x )= x b. f ( x )= c. f ( x )= x 2 1 x 4 2 9. For each of the functions below, indicate whether the function is one-to-one, onto or both. If the function is not one-to-one, give an example showing why. a. f : RR. f ( x )=x 2 b. g :R R . g ( x ) =x3 c. h :Z Z . h ( x )=x 3 10. Give the first ten terms of the sequence whose nth term is the largest integer k such that k !n. 11. Express the following sum using summation notation: (2 )+ (1 ) +0+1+2+3+ 4 . 12. The exclusive-or operator , is defined by the rule that a b is true whenever a or b is true but not both. Prove or disprove that x + (y z) = (x + y) (x + z). 13. Is the following function satisfiable? (x+ y)( + )( + z) y z x 14. Consider a school with four classes and two time slots during which the courses can be scheduled. The classes are named A, B, C, and D. The following pairs of courses can not be scheduled at the same time: (B, C), (A, D), (C, D), (B, A). Express the scheduling problem as a boolean expression. That is, give a boolean expression that is true if and only if there is a feasible schedule for the courses that satisfies all the constraints