1. Let A={ c,d,e,f }, B={f,j}, and C={d,g} , answer the following questions. Give reasons for your answers. a. Is B A? b. Is C A? c. Is C C? d. Is C a proper subset of A. For problems 2 through 6, let the universe be set ???? = {1,2,3,4,5,6,7,8,9,10}. Let ???? = {1,4,7,10}, ???? = {1,2,3,4,5}, and ???? = {2,4,6,8}. List the elements of each set. 2. ???? ???? 3. ???? 4. ???? ????

5. ???? 6. ???? ???? ???? 7. For the sets ???? = {1,2,3} and ???? = {????|???? > ???????????? ???? B } and ???? = {????|???? > ???????????? ???? B

9. Given that proposition ???? is false, proposition ???? is true, and proposition r is false. Determine if each of the propositions below are true or false. a. ???? ???? b. ????(???? ????) c. (????????) M(???? ????) (???? ????)N 10. Write the truth table for (???? ????) (???? ????)

11. Assuming that ???? and ???? are false and that ???? and ???? are true find the truth value for (???? ????)(???? ????). 12. Write the truth table for (???? ????)(???? ????). 13. Given the propositions ????: You run 10 laps daily. ????: You are healthy. ????: You take multi-vitamins. Write the following symbolically. If you do not run 10 laps daily or do not take multi-vitamins, then you will not be healthy.

14. Given the propositions ????: Today is Monday, ????: it is raining, ????: it is hot. Formulate the symbolic expression in words. (???? ????) ????

15. Given propositions ???? = (???? ????) (???? ????) and Q=(???? ????). Are P and Q logically equivalent? Prove your answer. 16. For the following argument written in words formulate a symbolic argument and then determine if the argument is valid. If I study hard, then I get A's. If I don't get rich then I don't get A's. Therefore I get rich.

For problems 17 and 18 write the given argument in words and determine whether each argument is valid. Let p: 4 megabytes is better than no memory at all. q: We will buy more memory. r: We will buy a new computer. 17. ???? (???? ????) ???? ???? ______ ???? ???? 18. ???? (???? ????) ???? (???? ????) ---------------- (???? ????) ????

19. Use truth tables to prove that a conditional statement is not logically equivalent to its inverse. 20. Use truth tables to determine if the following argument form is valid. Indicate which columns represent the premises and which represent the conclusion and include a sentence to explaining how the truth table supports your answer. p p q ~ q r ----------- r

21. Let P(x) be the predicate "x > 1/x." a. Write P(2), P(1/2), P(-1), P(-1/2), and P(8), and indicate which of these statements are true and which are false. b. Find the truth set of P(x) if the domain of x is R, the set of all real numbers. c. If the domain is the set > of all positive real numbers, what is the truth set of P(x).

22. Let Q(x,y) be the predicate "???????? ????

24. Each of the following statements is true. In each case write the converse of the statement, and give a counterexample showing that the converse is false. a. If n is any prime number that is greater than 2, then n+1 is even. b. If m is any positive integer, then 2m is even. c. If two circles intersect in two common points then they do not have a common center.

25. For each of the following equations, determine which of the following statements are true: i. For all real numbers x, there exists a real number y such that the equation is true. ii. There exist a real number x, such that for all real numbers y, the equation is true. a. 2???? + ???? = 7 b. ???? + ???? = ???? + ???? c. ???? B + 2???????? + ???? B d. (???? 5)(???? 1) = 0 e. ???? B + ???? B = 1