CMPSCI 190DM: A Mathematical Foundation for Informatics

Final Exam

David Mix Barrington

10 December 2013

Directions:

• Answer the problems on the exam pages.
• There are seven problems for 125 total points. Actual scale is A = 105, C = 75.
• If you need extra space use the back of a page.
• No books, notes, calculators, or collaboration.

  Q1: 20 points
  Q2: 15 points
  Q3: 15 points
  Q4: 20 points
  Q5: 15 points
  Q6: 20 points
  Q7: 20 points
Total: 125 points

• Question 1 (20): Briefly identify the following terms or concepts (4 points each):
  • (a) a function from a set A to a set B

  • (b) the inductive hypothesis of an induction proof

  • (c) the rule for counting the union of two finite sets A and B

  • (d) a connected component in an undirected graph

  • (e) the identity matrix

• Question 2 (15): Let p, q, r and s be four boolean variables. I am going to choose values for these variables, but all sets of values will not be equally likely. Instead I will choose one of the four variables to be true, with each of the four equally likely to be chosen, and set the other three variables to false.
  • (a, 5) What is the probability that the statement "p ∨ q ∨ (¬r)" will be true, given my values for the four variables?

  • (b, 10) What is the probability that the statement "(q ∨ r) → ¬(p → (r ∨ s))" will be true, given my values for the four variables?

• Question 3 (15): Let g(n) be the number of binary sequences of length n where the first three digits are all alike. (For example, g(0) = g(1) = g(2) = 0, and g(3) = 2 because the only possible sequences are 000 and 111.)

  Prove by induction on n, starting with either n = 3, that g(n) = 2^n-2.

• Question 4 (20): Suppose I throw a six-sided die three times, to get a sequence of three numbers each chosen from the set {1, 2, 3, 4, 5, 6}, with each such sequence being equally likely.
  • (a, 5) How many different three-number sequences are there? (Remember that you do not have to evaluate an arithmetic expression.)

  • (b, 5) How many of these sequences have all three numbers the same?

  • (c, 5) How many of these sequences have all three numbers different?

  • (d, 5) What is the probability that two of the three numbers are the same, and the third is different?

• Question 5 (15): Let f(n) be the probability that in n independent tosses of a fair coin, I get heads exactly once. So f(0) = 0 and f(1) = 1/2. We are going to derive a general formula for f(n). The probability that we get the first heads on the n+1'st toss is exactly 1/2ⁿ⁺¹. If we have one heads in the first n tosses, the probability that we do not get another on the n+1'st toss is 1/2.

  So we know that f(n+1) = (1/2)f(n) + 1/2ⁿ⁺¹. Using this fact, prove by induction on all positive integers n that f(n) = n/2ⁿ. (Use a base case of n = 1.)

• Question 6 (20): Let B be the following integer matrix:

  
          1  2
          2  0
        

  • (a, 5) Draw the directed multigraph whose matrix is B. Name the vertices x and y, where the rows and columns correspond to the two vertices in that order.

  • (b, 5) Compute the matrix B².

  • (c, 10) In the directed multigraph of part (a), how many three-step paths are there from vertex x to itself? Explain your answer.

• Question 7 (20): Caroline is a trombonist with the UMass Marching Band, who has been assigned extra marching practice. On the command "left face", she is supposed to turn 90 degrees to her left. She does this correctly 90% of the time, but the other 10% of the time she turns 90 degrees to the right instead. Her choice of turn on each command is independent of the others. She begins facing north, and gets a series of "left face" commands, so that at any later time she is facing either north, east, south, or west.
  • (a, 5) Write down the transition matrix T for the Markov process that describes Caroline's direction change after one command. Use the row order north-east-south-west.

  • (b, 5) Compute the matrix T². Note that many calculations are repeated.

  • (c, 5) If she begins facing north, what is the probability that she is facing south after two commands?

  • (d, 5) If she begins facing north, what is the probability that she is again facing north after four commands? Remember that you do not have to evaluate an arithmetic expression.

Last modified 12 December 2013