This is a custom exam written by Trevor, from TrevTutor.com that covers Set Theory,. Logic, and Counting. The questions
Discrete Mathematics 1 TrevTutor.com Midterm 1 Time Limit: 70 Minutes
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This exam contains 8 pages (including this cover page) and 9 questions. The total number of points is 70. This is a custom exam written by Trevor, from TrevTutor.com that covers Set Theory, Logic, and Counting. The questions are based off the videos on the website, as well as questions in the free textbook The Book of Proof. Please use this midterm as a guide for figuring out what your weak points are with the material. If you’d like to see more practice exams, as well as the video solution to this exam, check out TrevTutor.com.
Question Points Score 1
10
2
5
3
5
4
5
5
5
6
10
7
10
8
10
9
10
Total:
70
Discrete Mathematics 1
Midterm 1 - Page 2 of 8
1. (10 points) Suppose A = {a, b, c, d} and B = {1, 2}. Write out the following sets (a) (2 points) A × B
(b) (2 points) A ∪ B
(c) (2 points) A ∩ B
(d) (2 points) B − A
(e) (2 points) P(B)
Discrete Mathematics 1
Midterm 1 - Page 3 of 8
2. (5 points) Prove the following statement. (Hint: This is an equality, so you must prove that each set is a subset of the other. Use an arbitrary element x of each set to do so.) A∪B =A∩B
3. (5 points) Prove or disprove the following using two Venn Diagrams. A−B =A∩B
Discrete Mathematics 1
Midterm 1 - Page 4 of 8
4. (5 points) Determine which of the following are statements. You do not need to state whether they are true or false. You do not need to explain your answer. (a) (1 point) It is currently sunny outside.
(b) (1 point) It can’t possible be raining, can it?
(c) (1 point) Every even integer is of the form 2n + 1.
(d) (1 point) I hope you do well on this question.
(e) (1 point) It is a fact that anyone who says this question is not a statement will be correct.
5. (5 points) Prove that the following formulas are equivalent using a truth table. ¬P ↔ Q ⇔ (P → ¬Q) ∧ (¬Q → P )
Discrete Mathematics 1
Midterm 1 - Page 5 of 8
6. (10 points) Formalize the following argument, then determine the result using logical inferences. “Either Derek works or Avery does not work. If it is not true that both Avery works and Brandon does not work, then clearly Celina does not work. However, Derek does not work.” Does Celina work?
Discrete Mathematics 1
Midterm 1 - Page 6 of 8
7. (10 points) The following question relates to counting. (a) (2 points) Suppose we have a committee of 15 people. How many ways can we select a president, vice president, and secretary?
(b) (2 points) Suppose we have 150 people entering a lottery to win discrete mathematics textbooks. How many ways can we choose 5 winners?
(c) (2 points) Suppose we have a binary string of length 15. How many possible strings will have exactly 6 zeros?
(d) (2 points) Suppose we have the set A = {1, 2, 3, 4, 5, 6, 7}. How many ways can choose 3 numbers to form a decreasing sequence? (Hint: < 6, 4, 4 > is decreasing.)
(e) (2 points) How many 6-digit license plates can we make if the first and third digits have to be odd, and the fifth digit has to be even?
Discrete Mathematics 1
Midterm 1 - Page 7 of 8
8. (10 points) Suppose we have the word PARAPPATHERAPPA. (a) (2 points) How many different arrangements are there if all letters are distinct?
(b) (4 points) How many different arrangements are there if all similar letters are not distinct?
(c) (4 points) How many different arrangements are there if there are no consecutive vowels?
Discrete Mathematics 1
Midterm 1 - Page 8 of 8
9. (10 points) The following question relates to combinations with repetition. (a) (2 points) How many integer solutions are there to x1 + x2 + x3 + x4 + x5 = 17 where xi ≥ 0.
(b) (3 points) How many integer solutions are there to x1 + x2 + x3 + x4 + x5 = 17 where x1 , x2 ≥ 3 and x3 , x4 , x5 > 0.
(c) (2 points) Suppose I have 13 textbooks that I want to place on 3 shelves. How many ways can I arrange my textbooks if order does not matter?
(d) (3 points) Suppose now, I want at least two textbooks on each shelf. How many ways can I arrange my textbooks if order does not matter?
