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
1. ( 29 points) Let A = 4,6, B = 5,, C = 10, 12, 11,13,14,15, 12, {16}, D = e, π, 0,2, 100 207, 53 10 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
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.
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.
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. ( 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.)
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