Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
During the study of discrete mathematics, I found this course very informative and applicable.The main points in these lecture slides are:Logically Equivalent, Inverse of Converse, Contrapositive, Inference Rules, Tautology, Domain and Co-Domain, Bijective Function, Valid Arguments, Inverse Function, Power Set, Indirect Proof, Arbitrary Element
Typology: Exams
1 / 11
This page cannot be seen from the preview
Don't miss anything!
Name:
NetID:
Lecture Section:
Section Leader:
General Directions
Which of the following is logically equivalent to p → q?
a) q ∨ ¬p
b) the contrapositive of p → q
c) the inverse of the converse of p → q
d) all of the above
Answer: (d).
Which of the following is a negation of ∀x∀y[((x > 0) ∧ (y > 0)) → (x + y > 0)]?
a) ∃x∃y[(x > 0) ∧ (y > 0) ∧ (x + y ≤ 0)]
b) ∃x∃y[((x ≤ 0) ∨ (y ≤ 0)) ∧ (x + y > 0)]
c) ∀x∀y¬[((x > 0) ∧ (y > 0)) → (x + y > 0)]
d) ∃x∃y[(x ≤ 0) ∨ (y ≤ 0) ∨ (x + y > 0)]
Answer: (a).
c) ∀x(S(x) → L(x)), ¬S(a) ∴ ¬L(a)
d) ∃x(S(x) ∧ L(x)), S(a) ∴ L(a)
Answer: (a).
Let f (x) = 3x + 2 and g(x) = x^2 be functions defined on the integers (f : Z → Z, g : Z → Z). Which of the following is true?
a) g ◦ f = O(x^2 )
b) g ◦ f = O(x^3 ), and g ◦ f is not O(x^2 )
c) g ◦ f (x) = f ◦ g(x)
d) g ◦ f has an inverse function.
Answer: (a).
Which of the following is false?
a) {x} ⊆ {x}
b) {x} ∈ {x, {x}}
c) {x} ⊆ P({x}), where P({x}) is the power set of {x}
d) {x} ⊆ {x, {x}}
Answer: (c).
Use an indirect proof for the following:
Given: p → (m → w) w → d m ¬d Prove: ¬p
(7) is a contradiction of the given hypothesis (˜d). Since the negation of the conclusion led to a contradiction, the original conclusion (˜p) is proven.
Grading rubric:
2 pts. for assuming the opposite (p) 1 pt. for using "Given" statements 2,2 pts. for each modus ponens step (should have been used twice) 2 pts. for a correct conclusion (arriving at a contradiction) 1 pt. for labelling each step. Total: 10 pts.
If a student used a direct proof, we took off 3 pts.
Tell whether each of the following is True or False. The universe is all integers.
a) ∀z∀y∃x(x − y = z)
True
b) ∀y∃x∃z(x − y = z)
True
c) ∀x∀y∀z(x − y = z)
False. This is saying for all pairs of (x, y, z), it holds that x − y = z. Obviously, it is false.
d) ∀x∀y∃z(x − y = z)
True
e) ∀x∃y∃z(x − y = z)
True
f) ∃x∃y∀z(x − y = z)
False. This is saying, there exists a pair (x, y) such that x − y = z holds for any z. We know that for any fixed pair of (x, y), their difference is also fixed. Therefore, it is false.
g) ∃x∃y∃z(x − y = z)
True
h) ∃x∀y∀z(x − y = z)
False. It is saying there exists a particular x such that x − y = z holds for any pair of (y, z).
-1 point for each incorrect answer.
a) Let j and k be integers, with j even and k odd. Prove that the product of j and k is even.
j = 2n, and k = 2m + 1 for some integers n, m. Then jk = 2n(2m + 1) = 2z for some integer z. Thus, the product jk is even.
6 points for perfect answer. -2 points for not distinguishing m and n. -1 not concluding argument with the definition of an even number. -4 points for an answer that only specifies j and k, with no further argument. - points for an effort in the wrong direction.
b) Prove that the product of consecutive integers is even. Hint: you can use part a) in your solution.
Every pair of consecutive integers has one even and one odd. By part a) the product of such a pair is even.
7 points for perfect answer. -3 for restricting the least value to be even or odd. -1 point for other minor errors.
c) Prove that the square of an odd integer equals 8 k + 1 for some integer k. Hint: you can use part b) in your solution.
We wish to consider the square of an odd number: (2n + 1)^2. (2n + 1)^2 = 4n^2 + 4n + 1 = 4n(n + 1) + 1. Notice that from part b), n(n + 1) is even, so we have (2n + 1)^2 = 4(2k) + 1 for some integer k.
7 points for perfect answer. 2 points awarded for setting up problem and multiplying out the square without further argument. 4 points awarded for correct setup, multiplication, and factoring.
