Try It Now

Site: Saylor Academy
Course: CS202: Discrete Structures
Book: Try It Now
Printed by: Guest user
Date: Monday, November 28, 2022, 8:54 PM

Description

Work these exercises to see how well you understand this material.

Table of contents

Exercises

  1. If U = ℘ ({1, 2, 3, 4}), what are the truth sets of the following propositions?
    1. A ∩ {2, 4} = ∅.
    2. 3 ∈ A and 1 ∉ A.
    3. A ∪ {1} = A.
    4. A is a proper subset of {2, 3, 4}.
    5. |A| = |Ac|.

  2. Over the universe of positive integers, define:

    p
    (n): is prime and n < 32.
    q(n): n is a power of 3.
    r(n): n is a divisor of 27.
    1. What are the truth sets of these propositions?
    2. Which of the three propositions implies one of the others?

  3. If U = {0, 1, 2}, how many propositions over U could you list without listing two that are equivalent?

  4. Suppose that sis a proposition over {1,2, . . . , 8}. If Ts= {1,3, 5, 7}, give two examples of propositions that are equivalent to s.

  5. Let the universe be ℤ, the set of integers. Which of the following propositions are equivalent over Z?

    a
    : 0 < n2 < 9
    b: 0 < n3 < 27
    c: 0 < n < 3

 


Source: Al Doerr and Ken Levasseur, http://faculty.uml.edu/klevasseur/ads-latex/ads.pdf
Creative Commons License This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.

Solutions

  1. Answer:
    1. {{1}, {3}, {1, 3}, ∅}
    2. {{3}, {3, 4}, {3, 2}, {2, 3, 4}}
    3. {{1}, {1, 2}, {1, 3}, {1, 4}, {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {1, 2, 3, 4}}
    4. {{2}, {3}, {4}, {2, 3}, {2, 4}, {3, 4}}
    5. {AU : |A| = 2}

  2. Solution:

      1. Tp = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31}
      2. Tq = {1, 3, 9, 27, 81, . . . }
      3. Tr = {1, 3, 9, 27}
    1. r ⇒ q

  3. Answer: There are 23 = 8 subsets of U, allowing for the possibility of 28 nonequivalent propositions over U.

  4. Answer: Two possible answers: s is odd and (s − 1)(s − 3)(s − 5)(s − 7) = 0

  5. Solution: b and c