CMPSCI 190DM: A Mathematical Foundation for Informatics

Practice for Final Exam

David Mix Barrington

10 December 2015

Directions:

Answer the problems on the exam pages.
There are six problems for 125 total points. Scale will be determined after the exam but a good guess is A = 105, C = 70.
If you need extra space use the back of a page.
No books, notes, calculators, or collaboration.

  Q1: 15 points
  Q2: 15 points
  Q3: 25 points
  Q4: 30 points
  Q5: 15 points
  Q6: 25 points
Total: 100 points

Question 1 (15): Briefly explain the difference between the terms or concepts in each of the following pairs (3 points each):

(a) the inclusive or and exclusive or operations
(b) a counterexample and a contradiction
(c) a one-to-one function and an onto function
(d) the Product Rule and the Sum Rule for counting
(e) a directed graph and an undirected graph

Question 2 (15): Determine whether each of these five statements is true or false. No explanation is needed or wanted, and there is no penalty for guessing (3 points each).

(a) If p and q are any propositions, and q and (p → ¬q) are both true, then p must be false.
(b) If I prove P(1), and I prove that P(n) implies P(n+2) for any n, then P(n) must be true for all n.
(c) There are exactly 2³ = 8 ordered lists of length 2 from the set {a, b, c}.
(d) If I roll two six-sided dice, the probability that the sum of the numbers is in the set {7, 11} is exactly 2/11.
(e) If G is a directed graph with ten nodes and five edges, there must exist nodes x and y in G such that there is no path in G from x to y.

Question 3 (25): Vandals have sabotaged the traffic light in my neighborhood so that it now behaves randomly according to a Markov process. During every ten-second interval it is either red, yellow, or green. After an interval in which it is red, it is red in the following interval 90% of the time and green 10% of the time. After it is yellow, it is red 50% of the time and green 50% of the time. After it is green, it is green 90% of the time and yellow 10% of the time.

(a, 5) Write down the 3 by 3 matrix T of transition probabilities for this Markov process. Use the row order red-yellow-green.
(b, 5) Compute the matrix T².
(c, 5) If the light is now yellow, what is the probability that it is green 20 seconds (two time steps) later? Remember that you do not have to evaluate an arithmetic expression.
(d, 10) If the light is now red, what is the expected number of time steps that it will remain red before changing?

Question 4 (30): Let G be a directed graph with node set {A, B, C} and edge set {(A, B), (A, C), (B, A), (C, A)}.

(a, 5) Write down the adjacency matrix M for G, using the row order A-B-C.
(b, 5) Compute the matrices M² and M⁴.
(c, 5) How many four-step paths are there in G from node B to node C?
(d, 15) Prove, by induction on all positive numbers n, that there are exactly 2ⁿ paths of length 2n in G from node A to itself.

Question 5 (15): You are dealt three cards from a standard 52-card deck.

(a, 5) How many different possible three-card hands are there?
(b, 5) How many of these hands have three cards of the same suit?
(c, 5) How many of these hands have three cards of three different ranks?

Question 6 (25): Let c be the proposition "Cardie is hungry", d be the proposition "Duncan is asleep", and q be the proposition "both dogs are quiet".

(a, 5) Translate the statement "If both dogs are quiet, then Duncan must be asleep and Cardie is not hungry" into logical symbols.
(b, 5) Translate the statement "¬(¬q ∧ ¬c ∧ d)" into English.
(c, 15) With a truth table or otherwise, show that the statements of parts (a) and (b) together are equivalent to the statement "q ↔ ¬(d → c)". This said "q ↔ ¬(c → d)", which is wrong, until 13 December 2015.

Last modified 13 December 2015