CMPSCI 250: Introduction to Computation

Practice for First Midterm Exam

David Mix Barrington

28 February 2012

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 predicate

  • (b) a counterexample

  • (c) two logically equivalent statements

  • (d) a rational number

  • (e) the principle of mathematical induction

• Question 2 (20): Let p, q, and r be three propositional variables. Use a truth table to show that if p ∨ ¬q and q ∧ ¬r are both true, then p → r must be false.

• Question 3 (30): Prove by cases that if n is any integer, (n+1)(n+2)(n+3) is divisible by 6. (Hint: You don't necessarily have to multiply out the polynomial, though you may.)

• Question 4 (30): Suppose we want to use mathematical induction to prove that for every positive integer n, (n+1)(n+2)(n+3) is divisible by 6.
  • (a, 10) What is the base case?
  • (b, 10) What are the hypothesis and conclusion of the inductive case?
  • (c, 10) Prove the inductive case.

  Last modified 14 October 2013