My dear students, Believe in yourselves. Believe in your abilities. You have got this! -Dr. M

Transcription

1 1/20 2/10 3/7 4/18 5/10 6/6 7/17 8/12 Total/100 Please do not write in the spaces above. Directions: You have 50 minutes in which to complete this exam. You must show all work, or risk losing credit. Be sure to answer all questions asked. Good luck! J MATH Fall 2016 Dr. Morton Name: Exam I Version I My dear students, Believe in yourselves. Believe in your abilities. You have got this! -Dr. M

2 1. (20 points) Let A =, 4, B = 2,5,17, C = 1,1 11,12, D = 12,1,2, π, 4, E = 1, F = 1,17, G = {1,9}. The universal set here is U = R. Find the following, using the clearest notation possible. You do not have to show any work here, unless I tell you otherwise: a. A F = b. A D= c. F-E= d. D-F= e. C Z= f. P(G)= g. B E= h. (show a little work here:) B F= i. D = j. How many proper subsets are there of E? List all proper subsets of E:

3 2. (10 points) For each r R B, define the indexed collection of sets to be the half-open interval A C = 1, D C. No work is needed to be shown in this problem. a. Find A D, if it exists. If it does not exist, explain why. b. Find A E, if it exists. If it does not exist, explain why. c. Find A F, if it exists. If it does not exist, explain why. d. Find A (D/H)., if it exists. If it does not exist, explain why. e. Find C R I A C. f. Find A r R + r. 3. (7 points) Find one partition, S, and one collection T, of the set A={0,1,2,3,4,5, } which satisfies all three of the following conditions: a. S = 3 b. There is a collection, T S where T = 2 and P Q X = 4 c. There is no B S such that B = 2. Make sure you briefly explain how your partition S and collection T satisfy their needed requirements.

4 4. (18 points) Suppose that we have the two open sentences P(x): x is even; Q(x): x < 5 over the domain S=N. For which x S are the following true? To get full credit, you must show all work and give all possible scenarios which lead to the sentence being true. a. P(x) Q(x) b. [~P x ] Q(x)

5 5. (10 points) Logic: Short answer: a. Is the statement p ~q [(~p) q] a tautology, a contradiction, or neither? Fill in the truth table below and then tell me your conclusions. Note: Do not skip any needed columns. p T T F F q T F T F Does this represent (circle one): a tautology, a contradiction, or neither? b. Negate the following, using positive language wherever possible: At least four of my friends love sci-fi. 5 is a negative number. 6. (6 points) Precisely define the following: a. A is a subset of B b. A c. A B

6 7. (17 points) Sets: a. Find an indexed collection { C n } n of distinct sets (that is, no two sets are equal) satisfying the following two conditions: e `fd C` =, 16, 8, 4, 2,0,2,4,8,16, whereas e C` = 0 Make sure that you explicitly list out what your C 1 and C 2 are. `fd. b. Let R = 0,5,10,15,20,. Write R in set-builder notation. c. Carefully shade in the Venn diagram below to indicate the region (B A) C A B C d. Find a collection of three subsets A, B, C of {1,2,3,4} which satisfy all of the following conditions: A, B, and C all have the same cardinality 1 A C and 1 B C 2 belongs to A and C but not B. 3 is an element of an even number of A, B, and C. 4 is an element of an odd number of A, B, and C. A, B, and C are not equal. You do not have to show any work here.

7 8. (12 points) Let Q = φ, 0,1, 1, (0, ). For the following, answer yes/no (no explanation is needed) Is {0} Q? Is φ Q? Is 0,1 Q? What is Q here? b. Let A = {x R: x 2} and B = x Z: 3 < x 4. Right now, A is in set builder notation. Rewrite it using interval notation instead. Right now, B is in set builder notation. Rewrite it so that it lists out all the elements in B instead. Is the ordered pair (-2,5) an element of the set A B? Explain, briefly. Draw a careful graph of A B in the space below.

8 Scratch paper: This must be handed in with your exam. I will not grade anything on this page. Please put your name on this page if you have ripped it off the exam.