Try It Now

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| = |A^c|.
      
      
2. 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.
   
   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?
   
   
5. Suppose that sis a proposition over {1,2, . . . , 8}. If T_s= {1,3, 5, 7}, give two examples of propositions that are equivalent to s.
   
   
7. 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
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. {A ⊆ U : |A| = 2}
      
      
2. Solution:
   1. 
      
      1. T_p = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31}
      2. T_q = {1, 3, 9, 27, 81, . . . }
      3. T_r = {1, 3, 9, 27}
   2. r ⇒ q
      
      
3. Answer: There are 2³ = 8 subsets of U, allowing for the possibility of 2⁸ nonequivalent propositions over U.
   
   
5. Answer: Two possible answers: s is odd and (s − 1)(s − 3)(s − 5)(s − 7) = 0
   
   
7. Solution: b and c