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$$ \bigcap_{i = 1}^{n}(A \times B_i) = A \times \left(\bigcap_{i = 1}^{n}B_i\right) $$
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Page 435
**Exercise Set 6.3**
For each of 1-4 find a counterexample to show that the statement is false.
Assume all sets are subsets of a universal set $U$.
1. For all sets $A$, $B$, and $C$,
$$ (A \cup B) \cap C = A \cup (B \cap C) $$
2. For all sets $A$ and $B$, $(A \cup B)^c = A^c \cup B^c$.
3. For all sets $A$, $B$, and $C$, if $A \nsubseteq B$ and $B \nsubseteq C$ then
$A \nsubseteq C$.
4. For all sets $A$, $B$, and $C$, if $B \cup C \subseteq A$ then
$$ (A - B) \cap (A - C) = \emptyset $$
For each of 5-21 prove each statement that is true and find a counterexample for
each statement that is false. Assume all sets are subsets of a universal set
$U$.
5. For all sets $A$, $B$, and $C$,
$$ A - (B - C) = (A - B) - C $$
6. For all sets $A$ and $B$, $A \cap (A \cup B) = A$.
7. For all sets $A$, $B$, and $C$,
$$ (A - B) \cap (C - B) = A - (B \cup C) $$
8. For all sets $A$ and $B$, if $A^c \subseteq B$ then $A \cup B = U$.
9. For all sets $A$ ,$B$, and $C$, if $A \subseteq C$ and $B \subseteq C$ then
$A \cup B \subseteq C$.
10. For all sets $A$ and $B$, if $A \subseteq B$ then $A \cap B^c = \emptyset$.
11. For all sets $A$, $B$, and $C$, if $A \subseteq B$ then
$A \cap (B \cap C)^c = \emptyset$.
12. For all sets $A$, $B$, and $C$,
$$ A \cap (B - C) = (A \cap B) - (A \cap C) $$
13. For all sets $A$, $B$, and $C$,
$$ A \cup (B - C) = (A \cup B) - (A \cup C) $$
14. For all sets $A$, $B$, and $C$, if $A \cap C = B \cap C$ and
$A \cup C = B \cup C$, then $A = B$.
15. For all sets $A$, $B$, and $C$, $(A - B) \cup C \subseteq A \cup (C - B)$.
16. For all sets $A$ and $B$, if $A \cap B = \emptyset$ then
$A \times B = \emptyset$.
17. For all sets $A$ and $B$, if $A \subseteq B$ then
$\mathscr{P}(A) \subseteq \mathscr{P}(B)$.
18. For all sets $A$ and $B$,
$\mathscr{P}(A \cup B) \subseteq \mathscr{P}(A) \cup \mathscr{P}(B)$.
19. For all sets $A$ and $B$,
$\mathscr{P}(A) \cup \mathscr{P}(B) \subseteq \mathscr{P}(A \cup B)$.
20. For all sets $A$ and $B$,
$\mathscr{P}(A \cap B) = \mathscr{P}(A) \cap \mathscr{P}(B)$.
21. For all sets $A$ and $B$,
$\mathscr{P}(A \times B) = \mathscr{P}(A) \times \mathscr{P}(B)$.
22. Write a negation for each of the following statements. Indicate which is
true, the statement or its negation. Justify your answers.
a. $\forall$ sets $S$, $\exists$ a set $T$ such that $S \cap T = \emptyset$.
b. $\exists$ a set $S$ such that $\forall$ sets $T$, $S \cup T = \emptyset$.
23. Let $S =\{a, b, c\}$, and for each integer $i = 0, 1, 2, 3$, let $S_i$ be
the set of all subsets of $S$ that have $i$ elements. List the elements in
$S_0, S_1, S_2$, and $S_3$. Is $\{S_0, S_1, S_2, S_3\}$ a partition of
$\mathscr{P}(S)$?
24. Let $A = \{t, u, v, w\}$, and let $S_1$ be the set of all subsets of $A$
that do not contain $w$ and $S_2$ the set of all subsets of $A$ that contain
$w$.
a. Find $S_1$.
b. Find $S_2$.
c. Are $S_1$ and $S_2$ disjoint?
d. Compare the sizes of $S_1$ and $S_2$.
e. How many elements are in $S_1 \cup S_2$?
f. What is the relation between $S_1 \cup S_2$ and $\mathscr{P}(A)$?
25. Use mathematical induction to prove that for every integer $n \geq 2$, if a
set $S$ has $n$ elements, then the number of subsets of $S$ with an even
number of elements equals the number of subsets of $S$ with an odd number of
elements.
26. The following problem, devised by Ginger Bolton, appeared in the January
1989 issue of the _College Mathematics Journal_ (Vol. 20, No. 1, p. 68):
Given a positive integer $n \geq 2$, let $S$ be the set of all nonempty
subsets of $\{2, 3, \dots, n\}$. For each $S_i \in S$, let $P_i$ be the
product of the elements of $S_i$. Prove or disprove that
$$ \sum_{i = 1}^{]2^{n - 1} - 1}{P_i} = \frac{(n + 1)!}{2} - 1 $$
In 27 and 28 supply a reason for each step in the derivationl.l
27. For all sets $A$, $B$, and $C$,
$$ (A \cup B) \cap C = (A \cap C) \cup (B \cap C) $
_Proof:_
Suppose $A$, $B$, and $C$ are any sets. Then
$$ (A \cup B) \cap C = C \cup (A \cup B) $$
by __ (a) __
$$ = (C \cap A) \cup (C \cap B) $$
by __ (b) __
$$ = (A \cap C) \cup (B \cap C) $$
by __ \(c\) __
28. For all sets $A$, $B$, and $C$,
$$ (A \cup B) - (C - A) = A \cup (B - C) $$
_Proof:_
Suppose $A$, $B$, and $C$ are any sets. Then
$$ (A \cup B) - (C - A) = (A \cup B) \cap (C - A)^c $$
by __ (a) __
$$ = (A \cup B) \cap (C \cap A^c)^c $$
by __ (b) __
$$ = (A \cup B) \cap (A^c \cap C)^c $$
by __ \(c\) __
$$ = (A \cup B) \cap ((A^c)^c \cup C^c) $$
by __ (d) __
$$ = (A \cup B) \cap (A \cup C^c) $$
by __ (e) __
$$ = A \cup (B \cap C^c) $$
by __ (f) __
$$ = A \cup (B - C) $$
by __ (g) __
29. Some steps are missing from the following proof that for all sets $A$ and
$B$, $(A \cup B^c) - B = (A - B) \cup B^c$. Indicate what they are, and then
write the proof correctly.
**Proof:**
Let any sets $A$ and $B$ be given. Then
$$ (A \cup C^c) - B = (A \cup B^c) \cap B^c $$
by the set difference law
$$ = (B^c \cap A) \cup (B^c \cap B^c) $$
by the distributive law
$$ = (B^c \cap A) \cup B^c $$
by the idempotent law for $\cup$
$$ (A - B) \cup B^c $$
by the set difference law.
In 30-40, construct an algebraic proof for the given statement. Cite a property
from Theorem 6.2.2 for every step.
30. For all sets $A$, $B$, and $C$,
$$ (A \cap B) \cup C = (A \cup C) \cap (B \cup C) $$
31. For all sets $A$ and $B$, $A \cup (B - A) = A \cup B$.
32. For all sets $A$ and $B$, $(A - B) \cup (A \cap B) = A$.
33. Fora ll sets $A$ and $B$, $(A - B) \cap (A \cap B) = \emptyset$.
34. For all sets $A$, $B$, and $C$,
$$ (A - B) - C = A - (B \cup C) $$
35. For all sets $A$ and $B$, $A - (A - B) = A \cap B$.
36. For all sets $A$ and $B$, $((A^c \cup B^c) - A)^c = A$.
37. For all sets $A$ and $B$, $(B^c \cup (B^c - A))^c = B$.
38. For all sets $A$ and $B$, $(A \cap B)^c \cap A = A - B$.
39. For all sets $A$ and $B$,
$$ (A - B) \cup (B - A) = (A \cup B) - (A \cap B) $$
40. For all sets $A$, $B$, and $C$,
$$ (A - B) - (B - C) = A - B $$
In 41-43 simplify the given expression. Cite a property from Theorem 6.2.2 for
every step.
41. $A \cap ((B \cup A^c) \cap B^c)$
42. $(A - (A \cap B)) \cap (B - (A \cap B))$
43. $((A \cap (B \cup C)) \cap (A - B)) \cap (B \cup C^c)$
44. Consider the following set property: For all sets $A$ and $B$, $A - B$ and
$B$ are disjoint.
a. Use an element argument to derive the property.
b. Use an algebraic argument to derive the property (by applying properties from
Theorem 6.2.2).
c. Comment on which method you found easier.
35. Consider the following set property: For all sets $A$, $B$, and $C$,
$$ (A - B) \cup (B - C) = (A \cup B) - (B \cap C) $$
a. Use an element argument to derive the property.
b. Use an algebraic argument to derive the property (by applying properties from
Theorem 6.2.2).
c. Comment on which method you found easier.
**Definition:**
Given sets $A$ and $B$, the **symmetric difference of $A$ and $B$**, denoted
$A \Delta B$, is, is
$$ A \Delta B = (A - B) \cup (B - A) $$
46. Let $A = \{1, 2, 3, 4\}$, $B = \{3, 4, 5, 6\}$, and $C = \{5, 6, 7, 8\}$.
Find each of the following sets:
a. $A \Delta B$
b. $B \Delta C$
c. $A \Delta C$
d. $(A \Delta B) \Delta C$
Refer to the definition of symmetric difference given above. Prove each of
47-52, assuming that $A$, $B$, and $C$ are all subsets of a universal set $U$.
47. $A \Delta B = B \Delta A$
48. $A \Delta \emptyset = A$
49. $A \Delta A^c = U$
50. $A \Delta A = \emptyset$
51. If $A \Delta C = B \Dcelta C$, then $A = B$.
52. $(A \Delta B) \Delta C = A \Delta (B \Delta C)$
53. Derive the set identity $A \cup (A \cap B) = A$ from the properties listed
8n Theorem 6.2.2(1)-(9). Start by showing that for every subset $B$ of a
universal set $U$, $U \cup B = U$. Then intersect both sides with $A$ and
deduce the identity.
54. Derive the set identity $A \cap (A \cup B) = A$ from the properties listed
in Theorem 6.2.2(1)-(9). Start by showing that for every subset $B$ of a
universal set $U$, $\emptyset = \emptyset \cap B$. Then take the union of
both sides with $A$ and deduce the identity.