# Try It Now

 Site: Saylor Academy Course: CS202: Discrete Structures Book: Try It Now
 Printed by: Guest user Date: Sunday, July 21, 2024, 1:03 AM

## Description

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

## 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

## Solutions

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