Exam 1 February 3, 2015

Transcription

1 CSC Discrete Structures Exam 1 February 3, 2015 Name: KEY Question Value Score OAL 100 Please answer questions in the spaces provided. If you make a mistake or for some other reason need more space, please use the back of pages and clearly indicate where the answer can be found. Show your work carefully. Just writing an answer will not do. Show any assumptions, show the steps you took, and show how you came to your answer. Good luck!

2 1. [ /20] erminology For each of the following, write a brief but complete explanation of its meaning: Product principle Powerset Bipartite graph Z Contrapositive ree Bijection For each of the following graphs give the name and draw the graph for n = 4. Cn Kn Pn

3 2. [ /20] a) [ /5] When redeeming a prize coupon, you may choose one of six charms and" "either one of three carabiners or one of two bracelets. How many different prize choices could you make?" b) [ /5] Scary Clown offers a Sad Meal containing a sandwich, a salad, a dessert, and a drink. here are 5 types of sandwiches, 2 types of salads, and 4 different kinds of desserts. A person with low standards for food could eat a different Sad Meal every day for about 4 months! So, how many drinks are possible choices for a Sad Meal? Justify your answer. here are k = 165k choices and 1095 days, so k But, of course, k must be an integer... so there are at least seven possible drinks. c) [ /5] Is the product of two odd numbers even or odd? Prove it. d) [ /5] Perhaps keeping pigeons in mind, show that if a simple graph has at least two vertices, then two of its vertices must have the same degree. Proof by contradiction: Suppose all n vertices have different degrees. he smallest degree possible is 0, and the largest possible is n 1, which is a total of n different degrees. However, if a vertex has degree n 1, there cannot also be a vertex of degree 0. So, there are at most n 1 different degrees, and thus by the pigeonhole principle two of the n vertices must have the same degree.

4 3. [ /20] Compute the following: a) [ /2] {z Z z > 10, z 3 < 0} = hese are the negative integers { 9,..., 1}, so the cardinality of the set is 9. List the elements b) [ /3] P({ a, 0, {a, b}) = { c) [ /2] Express DeMorgan s law for the complement of a union: A B = A B = A B d) [ /3] Draw a Venn diagram representing (A B) (A C). d) [ /10] Mark the following statements as true or false. Explain briefly. " x Z (x+0 = x) " x Z (x 2 > x) $ x Z (x+x=x* x) " x Z $ y Z (x+y = 0) $ y Z " x Z (x+y = 0)

5 4. [ /20] a) [ /10] Use truth tables to show that (P Q) (P R) is equivalent to just (P R). P Q R P Q P R (P Q) (P R) F F F F F F F F F F F F F F F F F F F F F F b) [ /5] Draw the seven nonisomorphic subgraphs of K3. c) [ /5] Draw a graph with degree sequence 1, 1, 1, 2, 2, 5.

6 5. [ /20] Consider the following graphs: G 1 G 2 a) [ /5] Verify the handshaking lemma for G 1. d) [ /5] Which of the following are subgraphs of G 2? C 3 K 3,3 K 6 b) [ /5] Is either of the graphs bipartite? Justify your answer. c) [ /5] Are the graphs isomorphic? If so, exhibit the isomorphism. If not, find a property that should be preserved by isomorphism for which the two graphs differ.

7 d) [ /5] Which of the following are subgraphs of the graph on the right (G 2 )? C 3 K 3,3 K 6