CIS 375 Intro to Discrete Mathematics Exam 1 (Section M004: Blue) 6 October Points Possible

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1 Name: CIS 375 Intro to Discrete Mathematics Exam 1 (Section M004: Blue) 6 October 2016 Question Points Possible Points Received Total 100 Instructions: 1. This exam is a closed-book, closed-notes exam. 2. Legibility counts! Make sure I can read (and find!) your answers. If you need more room for an answer than that given, use the back side of the pages. Be sure to leave a note indicating where the answer is. 3. This test should have 8 pages (including this cover sheet). Let me know now if your copy does not have the correct number of pages. 4. Recall these sets: Z: set of all integers R: set of all real numbers Z + = {z Z : z > 0} Z = {z Z : z < 0} N= {z Z : z 0}

2 1. (6 points) Consider the following truth-table fragment, where A 1, A 2, A 3, A 4, A 5, A 6, A 7, A 8 all represent specific propositional logic formulas: A 1 A 2 A 3 A 4 A 5 A 6 A 7 A 8 F T T T F T F F F F F T F F T F T T F T F T T F F T T F T F F T T T T T T T T T T F F T T T T F F T T T F F F F F T T T F F T T (a) Give two formulas from the set {A 1, A 2, A 3, A 4, A 5, A 6, A 7, A 8 } that are logically equivalent to one another. (If none of them are logically equivalent to one another, then simply write none.) (b) List all of the formulas from the set {A 1, A 2, A 3, A 4, A 5, A 6 } that are logical consequences of A 8. (If none of them are logical consequences, then simply write none.) (c) List all of the formulas from the set {A 3, A 4, A 5, A 6, A 7, A 8 } that are logical consequences of A 1 A 2. (If none of them are logical consequences, then simply write none.)

3 2. (10 points) Construct a complete truth table for the following formula: (p (q r)) (q r)

4 3. (24 points) Suppose that sets Q, R, and S are defined as follows: Q = {6, 2, 1, 4, 9, 3} S = {2, 9, 7, 4, 5} T = {1, 7, 6, 4} Calculate the following: (a) {w + 10 : w S} (b) 2 {4,9} (c) S T (d) Q T (e) S T (f) {3, 5} {8, 1, 5} (g) {Q} (h) {{4, 5}, 16, 2, {9, 8}} S (i) S Q (j) T (k) T Q (l) 2 Q

5 4. (15 points) For each set requested below, give a nonempty set that satisfies the conditions. (No points will be given for empty sets.) (a) Give a set B such that {3, 4, 6} B = {1, 3, 6}. (b) Give a set Y such that {4, 8} Y and 2 Y = 4. (c) Give a nonempty set F such that F 2 {4,8,2} 2 {5,8}. (d) Give a set G such that G {1, 3, 5, 7} = G {1, 3, 5, 7} = 3. (e) Give nonempty sets J and M such that J M {(3, 24), (5, 18), (7, 24), (7, 19)}.

6 5. (15 points) Consider the following sets of animals: W : animals that have wings F : animals that fly T : animals that have toes E : animals that lay eggs S : animals that can sing C : animals that are cold-blooded (a) (9 points) Express each of the following English statements using the language of set theory: i. There are some animals with wings that neither fly nor lay eggs. ii. All animals that have either wings or toes (but not both) can sing. iii. No animal that has toes and doesn t fly can sing. (b) (6 points) Express each of the following set-theory statements in everyday (but unambiguous) English: i. S W C ii. E (T F )

7 6. (15 points) Consider the following definitions: An integer n is gammic provided there is an integer k such that n = 5k 2. An integer n is deltic provided there is an integer k such that n = 4k + 3. In addition, recall the following two facts about integers: Fact 1: The sum of any two integers is an integer. Fact 2: The product of any two integers is an integer. Give a direct proof of the following claim: Let a and b be integers. If a is gammic and b is deltic, then 9b + 4a is deltic. Note: As always, you should follow the course format for writing proofs. Be explicit about your intial assumptions, what you need to show, and your underlying reasoning. You do not need to rewrite the claim.

8 7. (15 points) Give a direct proof of the following claim: Let J, K, P, Q be sets. If J Q, then (J K) (J P ) Q (K P ). Note: As always, you should follow the course format for writing proofs. Be explicit about initial assumptions, what you need to show, and your underlying reasoning. You do not need to rewrite the claim.

CIS 375 Intro to Discrete Mathematics Exam 2 (Section M001: Yellow) 10 November Points Possible

Name: CIS 375 Intro to Discrete Mathematics Exam 2 (Section M001: Yellow) 10 November 2016 Question Points Possible Points Received 1 20 2 12 3 14 4 10 5 8 6 12 7 12 8 12 Total 100 Instructions: 1. This