CMPSCI 250: Introduction to Computation

Practice for Second Midterm Exam

David Mix Barrington

1 November 2013

Directions:

• Answer the problems on the exam pages.
• There are four problems for 100 total points. Scale will be determined after the exam but a good guess is 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: 20 points
  Q3: 30 points
  Q4: 30 points
Total: 100 points

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

  • (b) an anti-symmetric relation

  • (c) a partial order

  • (d) the Cartesian product A × B of two sets A and B

  • (e) the function C(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 {Biscuit, Cardie, Duncan, Nala}. Let R be the following binary relation on S: R = {(x, y): there is a letter of the alphabet that occurs in both dog x's name and in dog y's name}.
  • (a, 10) Draw the diagram of the relation R.

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

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

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

• Question 4 (25): This question concerns natural numbers in the set {100, 101, 102,..., 999}.
  • (a, 5) How many of these numbers are there?
  • (b, 10) How many of these numbers are divisible by 5? Explain your answer.
  • (c, 10) How many of the numbers counted in part (b) have their first and second digits different? Explain your answer.

  Last modified 6 November 2013