Discrete Mathematics Final Exam

Wednesday, June 11, 2008

Computer Science and Engineering, SNU

1. (10 points) Answer following questions.

a) Provide three simple formulas or rules that generate the terms of an integer sequence that begins with the list 1, 2, 4, 8, ...

b) Prove that there are an infinite number of formulas that can generate the integer  sequence starting with 1, 2, 4, 8. ...

2. (10 points) Show that

[eqn: R= sum _{i=0} ^{n} r ^{i} = {r ^{n+1} -1} over {r-1} ```````````````````````(r != 1)].

3. (20 points) Let [eqn: P(n)] be the statement that

[eqn: 1+ 1 over 4 + 1 over 9 + ... 1 over n^2 < 2- 1 over n], where n is an integer greater than 1.

a) What is the statement [eqn: P(2)], the basis step of the proof?

b) Show that [eqn: P(2)] is true.

c) What is the inductive hypothesis?

d) What do you need to prove in the inductive step?

e) Complete the inductive step.

f) Explain why these steps show that this inequality is true whenever n is an integer greater than 1.

4. (20 points) Give a recursive definition of the following.

a) The set of odd positive integers.

b) The set of positive integer powers of 3.

c) The set of polynomials with integer coefficients.

5. (20 points) A drawer contains a dozen brown socks and a dozen black socks. A man takes socks out at random in the dark.

a) How many socks must he take out to be sure that he has at least two socks of the same color?

b) How many socks must he take out to be sure that he has at least two black socks?

c) How many socks must he take out to be sure that he has at least three pairs of matching socks?

6. (20 points) Several alternative definitions of probability are possible. Explain the strengths and weaknesses of each of the followng definitions.

a) frequentist

b) Laplacian

c) Bayesian

d) axiomatic

7. (20 points) Answer the following questions on graphs.

a) Give a definition of a directed graph.

b) Define the indegree and the outdegree of a vertex v in a directed graph.

c) Show the relationship among the total number of indegrees, the total number of outdegrees, and the total number of edges in a directed graph.

d) Define the degree of a vertex v in an undirected graph.

e) Show the relationship between the total number of degrees and the total number of edges in an undirected graph.

8. (20 points) Answer the following questions on discrete structures.

a) Define a predicate as a relation.

b) Show that a function is a relation.

c) Explain the relationships between relation and graph.

d) What kind of graph is the WWW? Use the names of different types of graph.

(The End)

140 points in total