CMPSCI 190DM: A Mathematical Foundation for Informatics

Second Midterm Exam

David Mix Barrington

8 November 2013

Directions:

• Answer the problems on the exam pages.
• There are four problems for 100 total points. Actual scale was A = 90, C = 60.
• If you need extra space use the back of a page.
• No books, notes, calculators, or collaboration.

  Q1: 20 points
  Q2: 30 points
  Q3: 25 points
  Q4: 25 points
Total: 100 points

• Question 1 (20): Briefly identify the following terms or concepts (4 points each):
  • (a) a onto function

  • (b) an transitive relation

  • (c) an equivalence relation

  • (d) the power set of a set

  • (e) the function P(n, r)

• Question 2 (30): Let A, B, and C be any three sets. Assume that the statements "A ⊆ B" and "A ∩ C = ∅" are both true.
  • (a, 10) Draw a Venn diagram indicating the relationships among A, B, and C.

  • (b, 10) Consider the statement "(B ∩ C) - A = ∅". Can this statement be proved either true or false from the information given? If it can, do so.

  • (c, 10) Consider the statement "A - (B ∪ C) = ∅". Can this statement be proved either true or false from the information given? If it can, do so.

• Question 3 (25): Let S be the set of dogs {Ace, Biscuit, Cardie, Duncan, Ebony}. Let R be the following binary relation on S: R = {(x, y): dog x has fewer letters in his/her name than dog y does}.
  • (a, 10) Draw the diagram of the relation R.

  • (b, 5) Is R reflexive? Explain your answer.

  • (c, 5) Is R anti-symmetric? Explain your answer.

  • (d, 5) Is R transitive? Explain your answer.

• Question 4 (25): This question concerns three-letter words over the 26-letter alphabet {A, B, C,..., Z}. We consider all three-letter strings, not just those that are words in English. (Remember that an arithmetic expression need not be evaluated for a full-credit answer.)
  • (a, 5) How many three-letter words are there in all?

  • (b, 10) For this problem, a vowel is a member of the set {A, E, I, O, U} and a consonant is any letter that is not a vowel. How many of the three-letter words have consonants in the first and third positions and a vowel in the second position?

  • (c, 10) How many of the three-letter words counted in part (a) do not have the same letter twice (or more) in a row?

  Last modified 12 December 2013