CMPSCI 190DM: A Mathematical Foundation for Informatics

First Midterm Exam

David Mix Barrington

28 September 2015

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: 25 points
  Q4: 35 points
Total: 100 points

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

(a) the summation of a number sequence
(b) the inverse of an implication
(c) the Double Negative rule in propositional logic
(d) an existential quantifier
(e) for a compound proposition to be a tautology

Question 2 (20): For each of these two number sequences, give the next three numbers in the sequence, find a recursive formula, and find a closed formula. Don't forget that the recursive formula must specify the first element. (10 points each)

a₁ = -7, a₂ = -3, a₃ = 1, a₄ = 5, a₅ = 9
b₁ = -1, b₂ = 0, b₃ = 3, b₄ = 12, b₅ = 39

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 eats paper", and q means "Duncan barks too much". If x and y are dogs, the predicate S(x, y) means "dog x is sillier than dog y". All variables in quantifiers are of type "dog". (5 points each)

(a) (to symbols) It is not the case that if Cardie eats paper, then Duncan barks too much.
(b) (to English) ∀x: S(d, x) ∨ ¬S(x, c)
(c) (to symbols) If Cardie eats paper and Duncan barks too much, then both are sillier than all other dogs.
(d) (to English) S(d, c) ∨ (q ∧ S(c, d) ∧ ¬p)
(e) (to symbols) There is a dog that is sillier than every dog that is not sillier than either Cardie or Duncan.

Question 4 (35): Establish the following facts with truth tables.

(a, 15) The compound proposition "((p → ¬q) ∧ q) → ¬p)" is a tautology.
(b, 20) The compound propositions "(p ∧ ¬q ∧ ¬r) ∨ (q ∧ r)" and "(q ↔ r) ∧ (p ∨ q)" are logically equivalent. Remember that "↔" is the symbol for equivalence or "if and only if".

Last modified 13 October 2015