No.:
University Name:
Student of Toronto Faculty of Applied Science and Engineering Department of Electrical and Computer Engineering Midterm Examination, March 8, 2000 Second Year -- Programs 7 and 9
ECE 203S -- Discrete Mathematics
Examination Type: Closed Book Examiners: B. Bardakjian, F. Kschischang, S. Mirabbasi
Instructions
This examination paper consists of seven [7] pages (including this one). Please make sure that you have a complete paper. Write your name and student number in the space provided at the top of each page. Please enter the first letter of your last name in the box at the bottom left corner of this page. Answer all eleven [11] questions. The value of each question is noted on each page. A total of 100 marks is available. Answer each question directly on the examination paper, using the back of each page if necessary. Please indicate clearly where your work can be found. Show your work! Show all steps and present all results clearly. State any assumptions that you make. If we can't read it, we can't mark it. Please be neat, and use ink. Calculators are not permitted. Time: 2 hours.
Examiner's Report
Enter the first letter of
1. 2. 3. 4. 5. 6. 7.
/5 /5 /6 /10 /10 /10 /14
Name:
Student No.:
5 marks
1. Let A = {a, {b, c}, }. For each of the following statements, answer true or false. i. A ii. {b, c} A iii. |P (A)| = 16 iv. {b, c} P (A) v. {a} P (A)
5 marks
2. Use a truth table to prove or disprove the following logical implication: ((p q) ( p r)) (q r). Explain your reasoning. p T T T T F F F F q T T F F T T F F r T F T F T F T F
6 marks
3. Consider the following statement: "Every natural number can be written as the sum of the squares of two natural numbers." (a) State this sentence as a quantified predicate.
(b) What is the negation of this predicate? Write the negation in a form in which negations appear only within predicates (i.e., so that no negation is outside a quantifier or an expression involving logical connectives).
(c) State in words the negation of the statement.
Name:
Student No.:
10 marks
4. Using the rules of inference, what logical conclusion (of no more than 12 words), can be drawn from all of the following hypotheses: The (a) car doesn't have a flat tire only if we're lucky. (b) If the car has a flat tire, we're not going for a picnic. (c) The sun is shining if we're lucky. (d) We're going for a picnic if it is not raining. Give the details of your argument.
10 marks
5. A perfect number is a positive integer that is equal to the sum of its positive divisors, except itself. (For example, the positive divisors of 6 are 1, 2, 3, and 6. Therefore, since 1 + 2 + 3 = 6, the number 6 is perfect.) Prove the following theorem: "A perfect number is not a prime."
Name:
Student No.:
10 marks
6. Using mathematical induction, prove that for every positive integer n, (n + 4)! > 16(2n ).
Name: 7. Consider the function f : R - {4} R defined by f (x) =
4 marks
Student No.:
3x - 2 . x-4
(a) Is this function one-to-one? (Justify your answer.)
4 marks
(b) Is this function onto? (Justify your answer.)
3 marks
(c) Show that this function is O(1).
3 marks
(d) Show that this function is not o(1).
Name:
Student No.:
8. The digraph shown in the figure represents a relation R on the set {a, b, c, d}.
a b
d c
2 marks
(a) List the elements of R.
2 marks
(b) Give a 0-1 matrix representation for R.
4 marks
(c) Is R reflexive? If not, list the fewest possible number of elements that must be added to R to make R reflexive.
4 marks
(d) Is R symmetric? If not, list the fewest possible number of elements that must be added to R to make R symmetric.
4 marks
(e) Is R transitive? If not, list the fewest possible number of elements that must be added to R to make R transitive.
4 marks
(f) Is R an equivalence relation? If not, list the fewest possible number of elements that
Name:
Student No.:
5 marks
9. A weekend soccer league consists of n > 1 teams. Over the course of a season, suppose team i plays a total of g(i) games against other teams. Give a graph-theoretic argument that explains why
n
g(i)
i=1
is even.
5 marks
10. Suppose that all vertices in a simple graph G have degree k, where k is odd. Show that the total number of edges in G is a multiple of k.
11. How many edges do each of the following graphs contain?
2 marks
(a) The cycle Cn .
2 marks
(b) The complete graph Kn .