Midterm Exam Solutions: Counting, Combinatorics, Graph Theory, and Equivalence Relations, Exams of Discrete Mathematics

Download Midterm Exam Solutions: Counting, Combinatorics, Graph Theory, and Equivalence Relations and more Exams Discrete Mathematics in PDF only on Docsity! Midterm Exam 03. Thursday, April 26 Exercise 1. Formulate the basic counting techniques, namely, the product rule (4 pt), the sum rule (4 pt), and the inclusion-exclusion principle (4 pt). Exercise 2. A computer company receives 350 applications from computer graduates for a job planning a line of new Web servers. Suppose that 220 of these people majored in computer science, 147 majored in business, and 51 majored both in computer science and in business. How many of these applicant majored neither in computer science nor in business? (10 pt) In order to earn the full credit, clearly indicate each step of the solution of your problem and which counting techniques you used. Exercise 3. (14 pt) How many solutions are there to the equation x1 + x2 + x3 + x4 + x5 + x6 = 26 where xi, i = 1, 2, 3, 4, 5, 6, is a nonnegative integer such that x1 ≥ 1, x2 ≥ 2, x3 < 2, and x6 ≥ 5. Exercise 4. What is a transitive closure of a relation R? (4 pt) Let R be a relation on set A with |A| = 5 which is presented by matrix MR. Which matrix presents the transitive closure of R? (6 pt) Exercise 5. Construct Hasse diagram for the divisibility relation on the set {1, 2, 3, 6, 8, 12, 24, 36}. (6 pt) Find any compatible total order which is different from the “natural order” 1 < 2 < 3 < 4 < 6 < 8 < 12 < 24 < 36. (6 pt). Exercise 6. Define an equivalence relation (4 pt). Let A = P (Z) be the power set on the set of all integer numbers and relation R on A is defined by: (S, T ) ∈ R iff S and T have the same cardinality. Show that R is an equivalence relation. (1 pt) for listing the properties of the equivalence relation and (12 pt) for the rigorous proof that the indicated properties hold. Clearly indicate all equivalence classes. (6 pt) 1