CMPSCI 250: Introduction to Computation

First Midterm Exam

David Mix Barrington

7 October 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: 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) DeMorgan's Laws for propositions

  • (b) the contrapositive of an implication

  • (c) the Division Theorem

  • (d) an odd integer

  • (e) the conclusion of an implication

• Question 2 (20): Let p and q be two propositions. Prove that if the three premises "q → ¬p", "p ∨ q", and "¬(q ∧ ¬p)" are all true, then the statement "p → q" must be false.

  You may use a truth table, apply Proof by Cases, or use other argument patterns that we have learned. (The truth table is probably simplest.)

• Question 3 (30): Let n be any integer. Prove that n³ - n is divisible by 4 if and only if either n is odd or n is divisible by 4. (Hint: Use Proof by Cases based on applying the Division Theorem to n.)

• Question 4 (30): We define a sequence of numbers (a₁, a₂, a₃,...) by the following recurrence. We define a₁ to be 1 and for any k with k ≥ 2, we define a_k to be 2a_k-1 + k.

  We would like to prove by induction that for any positive integer n, this sequence obeys the closed formula a_n = 2ⁿ⁺¹ - n - 2.

  • (a, 10) What is the base case of the induction? Prove the base case.
  • (b, 10) In the inductive step of the proof, what do we assume and what do we intend to prove? State both the hypothesis and the conclusion carefully.
  • (c, 10) Complete the proof of the inductive step.

  Last modified 8 October 2014