Exam 1 Review F18 PDF

Preview

Summary

practice questions for exam 1 math 243...

Description

Math 243 – Discrete Mathematics

Exam 1 Review

The following list of problems is intended as a review of the concepts and problem types that we have covered in class up to this point. It is not intended as a list of sample exam problems. Since the exam is an individual exam, I recommend that you think about these problems on your own. Once you are finished, then discuss the solutions with your group. On every problem, explain your reasoning. 1. Provide counterexamples to the following propositions (i.e., examples that make the propositions false): (a) If the program compiles, then it will run perfectly. (b) if n is prime, then n is odd. (c) ∀x, y ∈ R, x2 + y2 > 0. (d) (p ∨ q) → r is logically equivalent to p → (q ∧ r). 2. True or False: ∃x ∈ R such that x2 = 2x. 3. Draw a truth table for (p → q) ∧ [(q ∧ ¬r) → (p ∨ r)]. Is this a tautology? 4. (a) True or False: If x ∈ R and x2 < 0 then x = 7. (b) Write (i) the converse, (ii) the inverse, and (iii) the contrapositive of the proposition in part (a). 5. Prove that n4 − n2 is divisible by 3 for all n ∈ N. 6. Prove that 21/3 is irrational. 7. Prove, for integers j and k, if j + k = 15, then j > 8 or k > 8. 8. Prove or disprove: The sum of three odd integers is always divisible by 3. 9. Prove that 5 does not divide n2 − 3 for integers n ≥ 1. 10. Prove or disprove: If 4 divides the product mn then 4 divides m or 4 divides n. 11. Prove or disprove: The sum of two primes is never prime. 12. Prove that (3, 5, 7) is the only ”prime triple” (set of three consecutive odd integers that are all prime).

1...