Discrete Mathematics Demystified

CHAPTER 3 Set Theory 49
Exercises
1. Let S ={1, 2, 3, 4, 5}, T ={3, 4, 5, 7, 8, 9}, U ={1, 2, 3, 4, 9}, V ={2, 4,
6, 8}. Calculate each of the following:
a. (S ∪ V ) ∩ U
b. (S ∩ T ) ∪ U
2. Let S be any set and let T =∅. What can you say about S × T ?
3. Prove the following formulas for arbitrary sets S, T , and U. [Hint: You may
find Venn diagrams useful to guide your thinking, but a Venn diagram is not
a proof.]
a. S ∩ (T ∪ U) = (S ∩ T ) ∪ (S ∩ U)
b. S ∪ (T ∩ U) = (S ∪ T ) ∩ (S ∪ U)
4. Draw Venn diagrams to illustrate parts (a) and (b) of Exercise 3.3.
5. Suppose that A ⊂ B ⊂ C. What is A\B? What is A\C? What is A ∪ B?
6. Describe the set Q\Z in words. Describe R\Q.
7. Describe Q × R in words. Describe Q × Z.
8. Describe (Q × R)\(Z × Q) in words.
9. Give an explicit description of the power set of S ={a, b, 1, 2}.
10. Let S ={a, b, c, d}, T ={1, 2, 3}, and U ={b, 2}. Which of the following
statements is true?
a. {a} ∈S
b. 1 ∈ T
c. {b, 2} ∈U
11. Write out the power set of each set:
a. {1, ∅, {a, b}}
b. {•, △,∂}
12. Prove using induction on k that if the set S has k elements and the set T has
l elements then the set S × T has k · l elements.