Dr. Jaffe
Advanced Mathematics
Review Sheet
NAME: _____________________________________________
1. Let U = {1, 2, 3, 4, 5} with A = {1, 2, 3}, B = {3, 4}, and C = {1, 2, 3, 4}.
a) Create a Venn Diagram
b) Which sets are subsets of each other?
2. Which of the following sets are equal? Explain.
a) {1, 2, 3}
b) {3, 2, 1, 3}
c) {3, 1, 2, 3}
d) {1, 2, 2, 3}
3.
a) If set A has 63 proper sets, what is |A|? Explain.
b) If a set B has 64 sets of odd cardinality, what is |B|? Explain.
4. Let A = {1, 2, 3, 4, 5, 7, 8, 10, 11, 14, 17, 18}. How many subsets of A contain 6 elements?
5. Give an example of three sets W, X, Y such that W ∈ X and X ∈ Y, but W ∉ Y.
6. When U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 2, 3, 4, 5}, B = {3, 4, 5, 6, 7}, and C = { 7, 8, 9}, find
a) A ∩ B
b) A ∪ B
c) B ∩ C
d) A ∪ C ∩ B
7. For U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, let A = {1, 2, 3, 4, 5}, B = {1, 2, 4, 8}, C = {1, 2, 3, 5, 7}, and
D = {2, 4, 6, 8}. Determine each of the following:
a) (A ∪ B) ∩ D
̅
b) 𝐶̅ ∪ 𝐷
c) A ∪ (B – C)
d) B – (C – D)
e) ̅̅̅̅̅̅̅̅
𝐶∩𝐷
f) A ∪ (B ∩ C)
8. Use a membership table to establish each of the following:
a) ̅̅̅̅̅̅̅
𝐴 ∩ 𝐵 = 𝐴̅ ∪ 𝐵̅
̅̅̅ ∩ 𝐶) = (𝐴 ∩ 𝐵̅) ∪ (𝐴̅ ∪ 𝐶̅ )
b) ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
(𝐴 ∩ 𝐵) ∪ (𝐴
9. Using the laws of set theory, simplify each of the following:
a) 𝐴 ∩ (𝐵 − 𝐴)
b) 𝐴̅ ∪ 𝐵̅ ∪ (𝐴 ∩ 𝐵 ∩ 𝐶̅ )