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

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1 1/29 2/22 3/12 4/8 5/9 6/20 otal/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 MAH 225 Spring 2017 Dr. Morton Name: Exam I Version I My dear students, Believe in yourselves. Believe in your abilities. You can DO this! -Dr. M

2 1. ( 29 points) Let A = 4,6, B = 5,, C = 10, 12, 11,13,14,15, 12, {16}, D = e, π, 0,2, , E =, 1, = 4, 5,6, G = x Q: x 4, H = 3,2 I = x Z x = 3y + 2 where y Z, J =, 3 [10, ) he universal set (for all but the sets C and E) is U = R. ind the following, using the clearest notation possible. You do not have to show any work here, unless I tell you otherwise: a. A B = b. A = c. -A= d. D G= e. D G= f. A Z= g. B C= h. P(H) = i. H= j. J= k. (show a little work here:) B A= l. E = m. he set I is in set builder notation. Carefully write I below by listing all of the elements in the set I. n. List all proper subsets of E: o. C = p. Is {10} C? Circle one: YES NO q. Is φ C? Circle one: YES NO r. Is 10, {16} C? Circle one: YES NO

3 2. (22 points) Suppose that we have the two open sentences P(x): x<2; Q(x): x = 0, 1, or 2 over the domain S=Z. (Note that in Q, that notation means absolute value.) a. ind the set for which P(x) is RUE (listing all elements): b. ind the set for which P(x) is ALSE (listing all elements): c. ind the set for which Q(x) is RUE (listing all elements): d. ind the set for which Q(x) is ALSE (listing all elements): e. ind the set for which P(x) Q(x) is RUE (listing all elements). o get full credit, you must show all work, giving explanations as needed, and give all possible scenarios which lead to the sentence being true. f. Write ~ P(x) using positive language: g. ind the set for which ~ P(x) is RUE (listing all elements): h. ind the set for which ~ P(x) is ALSE (listing all elements): i. ind the set for which [~P x ] Q(x) is RUE (listing all elements). o get full credit, you must show all work, giving explanations as needed, and give all possible scenarios which lead to the sentence being true.

4 3. (12 points) Sets: a. Carefully shade in the Venn diagram below to indicate the region (B C) A A B C b. ind 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 All four of the elements 1,2,3,4 appear in at least one of the sets A, B, C 1 A C A B = 2,3 B 2 is in an odd number of A, B, C 3 is an element of an even number of A, B, and C. A, B, and C are not equal. You do not have to show any work here. here is exactly one correct answer. c. Let A = {x R: x < 3} and B = [ 2, ) Right now, A is in set builder notation. Rewrite it using interval notation instead. Is the ordered pair (-2,6) an element of the set A B? Explain, briefly. Draw a careful graph of A B in the space below.

5 4. ( 8 points) or each r R`, define the indexed collection of sets to be the half-open interval A a = {0, r}. No work is needed to be shown in this problem. a. What is one universal set for A a here? b. ind A b, if it exists. If it does not exist, explain why. c. ind A c, if it exists. If it does not exist, explain why. d. ind A (b/e)., if it exists. If it does not exist, explain why. e. ind a R f A a, using the best possible notation. f. ind A r R + r, using the best possible notation. 5. (9 points) or each r +, define B a = x, y R e : y r. (Note: there are no conditions on x.) a. Carefully sketch B b below (if it exists). b. Carefully sketch B h below (if it exists). c. ind a R f B a and carefully sketch it below. d. ind a R f B a and carefully sketch it below. Please answer: My solution for the set (put in set notation): a R f B a is Please answer: My solution for the set (put in set notation): a R f B a is

6 6. ( 20 points) Short answer: a. Precisely define the following: o A is a subset of B o A B o P is a tautology. b. ind the truth table for the following, being careful to include all necessary columns. p ~q [(p q] p q c. Negate the following, using positive language wherever possible: At least eight of my friends love Math. b is a prime number. 4 is even and 8 is odd. a is positive or b is irrational. If m is a cat then n is a Dalek. (Note: If you do not know what a Dalek is, do not worry. It is a living creature that is all you need to know.)

7 Scratch paper: his 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.

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

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