Discrete Mathematics 1
Name:
TrevTutor.com
Final Exam
Time Limit: 180 Minutes
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.
Discrete Mathematics 1
Final Exam - Page 2 of 16
Question Points Score
1
10
2
6
3
10
4
10
5
10
6
10
7
15
8
7
9
8
10
8
11
10
12
5
13
5
14
7
15
8
16
8
17
5
Total:
142
Discrete Mathematics 1
1. (10 points) Let A; B and
Final Exam - Page 3 of 16
C
be subsets of all the integers, de ned by
A
= fxj0 < x < 5g
B
C
= f3; 7; 19; 25g
= 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.
Discrete Mathematics 1
Final Exam - Page 4 of 16
2. (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(
3. (10 points) Prove that [(p _ q ) ^ :p] ! q is a tautology with a truth table.
A
)?
Discrete Mathematics 1
Final Exam - Page 5 of 16
4. (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.
Discrete Mathematics 1
Final Exam - Page 6 of 16
5. (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?
Discrete Mathematics 1
Final Exam - Page 7 of 16
6. (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?
Discrete Mathematics 1
Final Exam - Page 8 of 16
7. (15 points) Solve the following parts.
(a) (5 points) Find the number of integer solutions to the equation x1 + x2 + x3 + x4 = 15
(b) (5 points) Find the number of integer solutions to the equation x1 + x2 + x3 + x4 = 15
where xi 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 x1 + x2 + x3 + x4 = 15.
Discrete Mathematics 1
Final Exam - Page 9 of 16
8. (7 points) Prove that if two integers have the same parity (odd or even), then their sum
is even.
Discrete Mathematics 1
Final Exam - Page 10 of 16
9. (8 points) Prove that if x5 + 7x3 + 5x x4 + x2 + 8, then x 0.
Discrete Mathematics 1
Final Exam - Page 11 of 16
10. (8 points) Suppose a; b; c 2 Z. Prove that if a2 + b2 = c2 , then
a
or b is even.
Discrete Mathematics 1
Final Exam - Page 12 of 16
11. (10 points) Using induction, prove that 21 + 22 + + 2n = 2n+1
2.
Discrete Mathematics 1
12. (5 points) Suppose
Is R symmetric? Is
Final Exam - Page 13 of 16
A
R
= fa; b; c; dg and R = f(a; a); (b; b); (c; c); (d; d)g. Is
transitive? If not, give supporting evidence.
R
re exive?
13. (5 points) Prove or disprove: If a relation is symmetric and re exive, it is also transitive.
Discrete Mathematics 1
Final Exam - Page 14 of 16
14. (7 points) Prove that the function
f
: R ! R de ned by f (x) = 5x3 + 1 is bijective.
Discrete Mathematics 1
Final Exam - Page 15 of 16
15. (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 5.
16. (8 points) Show that if
a; b; c;
and
d
are integers, such that ajc and bjd, then
j
ab cd
.
Discrete Mathematics 1
Final Exam - Page 16 of 16
17. (5 points) Using the Euclidian Algorithm, nd the greatest common denominator of
2002 and 2339.