Set 6.1: 12, 16 Set 6.2: 4, 10, 14 Set 6.3: 12, 37, 42
Logic Symbols: ≥ ≤ ≠ ¬ ∧ ∨ ⊕ ≡ → ↔ ∃ ∀ Set Symbols: ∈ ∉ ⊆ ⊂ ⊇ ⊃ ∅ ∪ ∩ ×
(6.1)
12. Let the universal set be the set R for all real numbers and let:
A = {x ∈ R | -3 ≤ x ≤ 0}
B = {x ∈ R | -1 < x < 2}
C = {x ∈ R | 6 < x ≤ 8}.
Find each of the following:
a. A ∪ B b. A ∩ B c. A^c d. A ∪ C
e. A ∩ C f. B^c g. A^c ∩ B^c h. A^c ∪ B^c
i. (A ∩ B)^c j. (A ∪ B)^c
16. Let A = {a, b, c}
B = {b, c ,d}
C = {b, c, e}
a. Find A ∪ (B ∩ C), (A ∪ B) ∩ C and (A ∪ B) ∩ (A ∪ C). Which of these sets are equal?
b. Find A ∩ (B ∪ C), (A ∩ B) ∪ C and (A ∩ B) ∪ (A ∩ C). Which of these sets are equal?
c. Find (A - B) - C and A - (B - c). Are these sets equal?
(6.2)
4. The following is a proof that for all sets A and B, if A ⊆ B, then A ∪ B ⊆ B. Fill in the blanks.
Proof: Suppose A and B are any sets and A ⊆ B. [We nust show that (a)]. Let x ∈ (b). [We must show that (c).] By definition of union, x ∈ (d) (e) x ∈ (f). In case x ∈ (g), then since A ⊆ B, x ∈ (h). In case x ∈ B, then clearly x ∈ B. So in either case, x ∈ (i) [as was to be shown.]
Use an element argument to prove each statement in 7-19. Assume that all sets are subsets of a universal set U.
10. For all sets A, B, and C,
(A - B) ∩ (C - B) = (A ∩ C) - B
(A - B) ∩ (C - B) ⊆ (A ∩ C) - B
(A ∩ C) - B ⊆ (A - B) ∩ (C - B)
