CMPSCI 190DM: A Mathematical Foundation for Informatics

Practice for First Midterm Exam

David Mix Barrington

20 September 2015

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: 15 points
  Q2: 30 points
  Q3: 25 points
  Q4: 30 points
Total: 100 points

Question 1 (20): Briefly identify the following terms or concepts (3 points each):
- (a) the converse of an implication
- (b) the contrapositive of an implication
- (c) an existential quantifier
- (d) the DeMorgan rules for ∧ and ∨
- (e) a contradiction (as a kind of compound proposition)
Question 2 (30): For each of these three number sequences, give the next three numbers in the sequence, find a recursive formula, and find a closed formula. (10 points each)
- a₁ = 7, a₂ = 2, a₃ = -7, a₄ = -12, a₅ = -17
- b₁ = 3, b₂ = 0, b₃ = -1, b₄ = 0, b₅ = 3
- c₁ = 3, c₂ = 4, c₃ = 6, c₄ = 10, c₅ = 18
Question 3 (25): Translate the following statements as indicated. The dogs Cardie and Duncan are denoted symbolically by c and d respectively. The proposition p means "Cardie is soft", and q means "Duncan is fierce". If x and y are dogs, the predicate S(x, y) means "dog x is smaller than dog y". (5 points each)
- (a) (to symbols) Cardie is soft, and if Duncan is fierce then Cardie is not soft.
- (b) (to English) p → ∃x: S(d, x)
- (c) (to symbols) Every dog is smaller than some dog that is larger than Duncan.
- (d) (to English) (p ⊕ (q ∧ S(c, d))) → ∀x: S(x, d)
- (e) (to symbols) Cardie is either soft or smaller than every dog except Duncan, or both.
Question 4 (30): Establish the following facts with truth tables.
- (a, 15) The compound proposition "p → (q → p)" is a tautology.
- (b, 15) The compound propositions "p ∧ (q ∨ ¬r)" and "(p ∨ q) ∧ (¬q ∨ p) ∧ (q ∨ ¬r)" are logically equivalent.

Last modified 22 September 2015