# Try It Now

Site: | Saylor Academy |

Course: | CS202: Discrete Structures |

Book: | Try It Now |

Printed by: | Guest user |

Date: | Thursday, November 30, 2023, 1:12 PM |

## Description

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

## Exercises

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

- Over the universe of positive integers, define:
(

p*n*):*n*is prime and*n*< 32.*q*(*n*):*n*is a power of 3.*r*(*n*): n is a divisor of 27.

- What are the truth sets of these propositions?
- Which of the three propositions implies one of the others?

- If
*U*= {0, 1, 2}, how many propositions over U could you list without listing two that are equivalent? - Suppose that
*s*is a proposition over {1,2, . . . , 8}. If*T*= {1,3, 5, 7}, give two examples of propositions that are equivalent to_{s}*s*. - Let the universe be ℤ, the set of integers. Which of the following propositions are equivalent over Z?
: 0 <

a*n*^{2}< 9*b*: 0 <*n*^{3}< 27*c*: 0 <*n*< 3

Source: Al Doerr and Ken Levasseur, http://faculty.uml.edu/klevasseur/ads-latex/ads.pdf

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.

## Solutions

- Answer:

- {{1}, {3}, {1, 3}, ∅}
- {{3}, {3, 4}, {3, 2}, {2, 3, 4}}
- {{1}, {1, 2}, {1, 3}, {1, 4}, {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {1, 2, 3, 4}}
- {{2}, {3}, {4}, {2, 3}, {2, 4}, {3, 4}}
- {
*A*⊆*U*: |*A*| = 2}

- Solution:

*T*= {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31}_{p}*T*= {1, 3, 9, 27, 81, . . . }_{q}*T*= {1, 3, 9, 27}_{r}

- r ⇒ q

- Answer: There are 2
^{3}= 8 subsets of*U*, allowing for the possibility of 2^{8}nonequivalent propositions over*U*. - Answer: Two possible answers:
*s*is odd and (*s*− 1)(*s*− 3)(*s*− 5)(*s*− 7) = 0 - Solution:
*b*and*c*