Discrete Math Homework
HOMEWORK Instructions: 1. You must show all of your work STEP-BY-STEP. 2. Explanations must be in complete sentences. ----------------------------------------------------------------- 1. Determine whether p→(q →r) and p→(q∧r) are equivalent using a truth table. 2. Prove that p→q and its converse are not logically equivalent using truth tables. 3. Write the contrapositive, converse, and inverse of the following: If you study for the test, then you will do well. 4. A set of propositions is consistent if there is an assignment of truth values to each of the variables in the propositions that makes each proposition true. Is the following set of propositions consistent? Write each statement in propositional logic and explain how you know if the system is consistent or not. (In other words, explain your answer.) r: The system is operating. s: The hard drive is functioning. t: The power source is connected. 1. If the system is operating, then the hard drive is functioning. 2. The hard drive is functioning if and only if the power source is connected. 3. The power source is not connected. 4. The system is operating or the power source is connected. 5. Suppose that P(x, y)= x+2y=xy. Determine the truth value of each statement. Show your work. a. ∃yP(3,y) b. ∀x∃yP(x,y) c. P(0,0) 6. Consider the following theorem: If x is an odd integer, then x +2 is odd. Give a proof by contraposition of this theorem. 7. Prove: if m and n are even integers, then mn is a multiple of 4. 8. Use a Venn diagram to determine which relationship,⊆, =, or ⊇, is true for the pair of sets. (In other words, draw two Venn diagrams). a. (A−B)∪(A−C), A−(B∩C) b. (A−C)−(B−C), A−B. 9. Suppose f:Z→Z has the rule f(n) = 3n−1. Determine whether f is onto Z. Prove how you know. 10. Let f(x) =⌊?3/3⌋. Find f(S) if S (write the answer as a set} is: a. {−2,−1,0,1,2,3} b. {0,1,2,3,4,5} 11. Verify that ?? = 6 is a solution to the recurrence relation ?? = 4??−1 −3??−2. Show your work. 12. Find ∑ ?2 + 2?8?=3 . Show your work. 13. Define the following sets S, T and U: S={1, 2, 3, 4, 5, 6, 7}, T={red, green, blue, yellow, purple, black, white}, U={a, b, c, d, e, f} The functions ?: ? → ? and ?: ? → ? are defined as follows: f: (1, green), (2, blue), (3, purple), (4, white), (5, black), (6, red), (7, yellow) g: (red, a), (green, d), (blue, a), (yellow, c), (purple, c), (black, f), (white, e) Define each of the following using the same form as above (as duples), if the function is not defined, explain why not. a. ?−1 b. ?−1 c. ? ∘ ? d. ? ∘ ?