Download Discrete Mathematics Midterm Practice: Sets, Logic, and Proofs and more Exercises Discrete Mathematics in PDF only on Docsity!
MATH 2080U / CSCI 2110U Discrete Mathematics
MIDTERM PRACTICE
Instructor: Dr. David
May 15, 2025
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Student Signature Student Number
Name of your TA Zahra H.
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Instructions:
- Use a pen to fill in the front page. If a midterm is written in pencil, you may not be allowed to re-submit questions for remarking.
- A non-programmable, non-graphing calculator is al- lowed. Laptops, and cellphones are NOT permitted.
- Before starting, read over the entire test carefully.
- Please verify that the test has 7 pages.
- Have your student card on your desk.
- There are 4 questions and a total of 0 marks. There are 85 minutes for the test.
- Questions do not carry equal weight so use your time wisely.
- Show as much work as needed to fully answer the ques- tions.
- You are expected to comply with OntarioTechU’s rules for academic conduct.
Question Points Score
1 0
2 0
3 0
4 0
Total: 0
- (a) Use set operations or membership table to show that A ∪ (B \ C) = (A ∪ B) ∩ (A ∪ C). (That is, ∀x(x ∈ A ∪ (B \ C) ↔ x ∈ (A ∪ B) ∩ (A ∪ C)), where x is in the universal set, U.)
(b) Let f : A → B, and let S and T be subsets of A. Show that f (S ∪ T) = f (S) ∪ f (T).
- (a) Using the rules of inference, show that the premises: (p ∧ q) ∨ r and r → s lead to the conclusion: p ∨ s. (State the reason for each step of the argument on the right.)
(b) The following argument is valid. True or False?
((p → q) ∧ ¬p) → ¬q.
- Multiple-Choice Questions.
(a) Consider the set {x ∈ R |x^2 + 2 x ≤ 3 }, find the values of x that satisfy the inequality x^2 + 2 x ≤ 3. A. [−3, 1] B. [−1, 4] C. {−1, 0, 2, 3, 4} D. {−3, −2, −1, 0, 1} (b) Let N and Z be set of natural numbers and integers respectively. Find Z \ Z −. A. Z + B. R C. Q D. N
(c) Consider the inveritble function, g : A → B, and an identity function, iA : A → A, such that iA(x) = x, ∀x ∈ A. Find the composition g ◦ g−^1. A. iA : A → A B. gA : A → A C. gB : B → A D. iB : B → B
(d) Let S = {−1, 0, 2, 5, 6} and f (x) = ⌊x/5⌋. Find f (S). A. { 0 } B. {0, 1, 2} C. {−1, 0, 1} D. { 1 }
(e) Find
p∈S
( 2 p + 1 ),
where S = {p ∈ R |p is odd and p ≤ 10 }. A. 38 B. 13, 923 C. 210 D. 55
(f) What is the complexity of the matrix-vector product of a n × n matrix and a n × 1 vector? A. n^2 B. n C. n^3 D. n!
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