The grade table is on the following page.

Transcription

1 Discrete Mathematics 1 TrevTutor.com Final Exam Time Limit: 180 Minutes Name: Class Section This exam contains 16 pages (including this cover page) and 17 questions. The total number of points is 142. This is a custom exam written by Trevor, from TrevTutor.com, that covers all of the material shown in the course. A lot of these questions are fairly straight-forward and are designed to show that the basic concepts and ideas of each section are understood. Some of these questions are also going to be challenging. Consider this nal exam as a method of nding general concepts that you have a weakness with, as opposed to unique situations that you don't quite understand. 100% on this exam will likely not be an A at your university. If you'd like to see more practice exams, as well as the video solution to this exam, check out TrevTutor.com. The grade table is on the following page.

2 Discrete Mathematics 1 Final Exam - Page 2 of 16 Question Points Score Total: 142

3 Discrete Mathematics 1 Final Exam - Page 3 of (10 points) Let A; B and C be subsets of all the integers, dened by A = fxj0 < x < 5g B = f3; 7; 19; 25g C = f1; 3; 5; 7; 11; 13; 17; 19; 23; 29g (a) (2 points) What is A [ B? (b) (2 points) What is B \ C? (c) (2 points) What is A B? (d) (2 points) What is A [ (B C)? (e) (2 points) Draw a Venn diagram of the sets A; B; C and label the elements in each set.

4 Discrete Mathematics 1 Final Exam - Page 4 of (6 points) Consider the set A = f?; a; bg (a) (2 points) What is jaj? (b) (2 points) What is P (A)? (c) (2 points) Is A P (A)? If not, what relationship does A have with P (A)? 3. (10 points) Prove that [(p _ q) ^ :p]! q is a tautology with a truth table.

5 Discrete Mathematics 1 Final Exam - Page 5 of (10 points) Consider the argument: "Mary is a diabetic. If Mary is a diabetic, then Frank is a television watcher. If Frank is a television watcher, then Mark is not unhappy. Either James is a watermelon, or Mark is happy." Formalize this argument, and make a conclusion about James.

6 Discrete Mathematics 1 Final Exam - Page 6 of (10 points) Consider selecting 4 objects from the set A = f1; 2; 3; 4; 5; 6; 9; 10; 12; 14; 15g. Evaluate the answers to each of the following questions. (a) (2 points) How many ordered sequences can be chosen from A without repetition? (b) (2 points) How many ordered sequences with repetition can be chosen from A? (c) (2 points) How many unordered sequences without repetition can be chosen from A? (d) (2 points) How many unordered sequences with repetition can be chosen from A? (e) (2 points) How many strictly decreasing sequences can be chosen from A?

7 Discrete Mathematics 1 Final Exam - Page 7 of (10 points) Consider the word UNUSUAL (a) (2 points) In how many ways can the letters in UNUSUAL be arranged? (b) (3 points) How many arrangements in part (a) have all three U's together? (c) (3 points) How many arrangements in part (a) have no consecutive U's? (d) (2 points) How many ways can the letters in UNUSUAL be arranged if every letter is distinct?

8 Discrete Mathematics 1 Final Exam - Page 8 of (15 points) Solve the following parts. (a) (5 points) Find the number of integer solutions to the equation x 1 +x 2 +x 3 +x 4 = 15 (b) (5 points) Find the number of integer solutions to the equation x 1 +x 2 +x 3 +x 4 = 15 where x i 1 for i = 1; 2; 3; 4 (c) (5 points) Create a problem that has the same meaning as nding the number of integer solutions to the equation x 1 + x 2 + x 3 + x 4 = 15.

9 Discrete Mathematics 1 Final Exam - Page 9 of (7 points) Prove that if two integers have the same parity (odd or even), then their sum is even.

10 Discrete Mathematics 1 Final Exam - Page 10 of (8 points) Prove that if x 5 + 7x 3 + 5x x 4 + x 2 + 8, then x 0.

11 Discrete Mathematics 1 Final Exam - Page 11 of (8 points) Suppose a; b; c 2 Z. Prove that if a 2 + b 2 = c 2, then a or b is even.

12 Discrete Mathematics 1 Final Exam - Page 12 of (10 points) Using induction, prove that n = 2 n+1 2.

13 Discrete Mathematics 1 Final Exam - Page 13 of (5 points) Suppose A = fa; b; c; dg and R = f(a; a); (b; b); (c; c); (d; d)g. Is R reexive? Is R symmetric? Is R transitive? If not, give supporting evidence. 13. (5 points) Prove or disprove: If a relation is symmetric and reexive, it is also transitive.

14 Discrete Mathematics 1 Final Exam - Page 14 of (7 points) Prove that the function f : R! R dened by f(x) = 5x is bijective.

15 Discrete Mathematics 1 Final Exam - Page 15 of (8 points) Prove that if six numbers are chosen at random, then at least two of them will have the same remainder when divided by (8 points) Show that if a; b; c; and d are integers, such that ajc and bjd, then abjcd.

16 Discrete Mathematics 1 Final Exam - Page 16 of (5 points) Using the Euclidian Algorithm, nd the greatest common denominator of 2002 and 2339.