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given set that is represented as the array $a[1], a[2], \dots, a[n]$.
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Page 427
**Exercise Set 6.2**
1.
a. To say that an element is in $A \cap (B \cup C)$ means that it is in __ (1)
__ and in __ (2) __.
b. To say that an element is in $(A \cap B) \cup C$ means that it is in __ (1)
__ or in __ (2) __.
c. To say that an element is in $A - (B \cap C)$ means that it is in __ (1) __
and not in __ (2)__.
d. To prove that $(A \cup B) \cap C \subseteq A \cup (B \cap C)$, we suppose
that $x$ is any element in __ (1) __. Then we must show that __ (2) __.
e. If $A$, $B$, and $C$ are any sets such that $B \subseteq C$, to prove that
$A \cap B \subseteq A \cap C$, we suppose that $x$ is any element in __ (1) __.
Then we must show that __ (2) __.
2. The following are two proofs that for all sets $A$ and $B$,
$A - B \subseteq A$. The first is less formal, and the second is more formal.
Fill in the blanks.
a. **Proof:** Suppose $A$ and $B$ are any sets. To show that
$A - B \subseteq A$, we must show that every element in __ (1) __ is in __ (2)
__. But any element in $A - B$ is in __ (3) __ and not in __ (4) __ (by
definition of $A - B$). In particular, such an element is in $A$.
b. **Proof:** Suppose $A$ and $B$ are any sets and $x \in A - B$. _[We must show
that __ (1) __.]_ By definition of set difference, $x \in$ __ ( 2 ) __ and
$x \notin$ __ (3) __. In particular, $x \in$ __ (4) __ _[which is what was to be
shown]._
In 3 and 4, supply explanations of the stesp in the given proofs.
3. **Theorem:** For all sets $A$, $B$, and $C$, if $A \subseteq C$,
$B \subseteq C$, then $A \subseteq C$.
**Proof:**
| Statement | Explanation |
| ------------------------------------------------------------------------------------ | ------------------------------------- |
| Suppose $A$, $B$, and $C$ are any sets such that $A \subseteq B$ and $B \subseteq C$ | starting point |
| We must show that $A \subseteq C$. | conclusion to be shown |
| Let $x$ be any element in $A$. | start of an element proof |
| Then $x$ is in $B$. | __ (a) __ |
| It follows that $x$ is in $C$. | __ (b) __ |
| Thus every element in $A$ is in $C$ | since $x$ could be any element of $A$ |
| Therefore, $A \subseteq C$ _[as was to be shown]._ | __ \(c\) __ |
4. **Theorem:** For all sets $A$ and $B$, if $A \subseteq B$, then
$A \cup B \subseteq B$.
**Proof:**
| Statement | Explanation |
| ----------------------------------------------------------------- | -------------------------------------------- |
| Suppose $A$, $B$, and $C$ are any sets such that $A \subseteq B$. | starting point |
| We must show that $A \cup B \subseteq B$ | conclusion to be shown |
| Let $x$ be any element in $A \cup B$. | start of an element proof |
| Then $x$ is in $A$ or $x$ is in $B$. | __ (a) __ |
| In case $x$ is in $A$, then $x$ is in $B$ | __ (b) __ |
| In case $x$ is in $B$, then $x$ is in $B$. | tautology ($p \to p$) |
| So in either case $x$ is in $B$. | proof by division into cases |
| Thus every element in $A \cup B$ is in $B$ | since $x$ could be any element of $A \cup B$ |
| Therefore, $A \cup B \subseteq B$ _[as was to be shown]._ | __ \(c\) __ |
5. Prove that for all sets $A$ and $B$, $(B - A) = B \cap A^c$.
6. Let $\cap$ and $\cup$ stand for the words "intersection" and "union",
respectively. Fill in the blanks in the following proof that for all sets
$A$, $B$, and $C$, $A \cap (B \cup C) = (A \cap C) \cup (A \cap C)$.
**Proof:** Suppose $A$, $B$, and $C$ are any sets.
(1) Proof that $A \cap (B \cup C) \subseteq (A \cap B) \cup (A \cap C)$:
Let $x \in A \cap (B \cup C)$. _[We must show that $x \in$ __ (a) __ ]._
By definition of $\cap$, $x \in$ __ (b) __ and $x \in B \cup C$.
Thus $x \in A$ and, by definition of $\cup$, $x \in B$ or __ \(c\) __.
_Case 1 $(x \in A \text{ and } x \in B)$:_ In this case, $x \in A \cap B$ by
definition of $\cap$.
_Case 2 $(x \in A \text{ and } x \in C)$:_ IN this case, $x \in A \cap C$ by
definition of $\cap$.
By cases 1 and 2, $x \in A \cap B$ or $x \in A \cap C$, and so, by definition of
$\cup$, __ (d) __.
_[So $A \cap (B \cup C) \subseteq (A \cap B) \cup (A \cap C)$ by definition of
subset.]_
(2) Proof that $(A \cap B) \cup (A \cap C) \subseteq A \cap (B \cup C)$:
Let $x \in (A \cap B) \cup (A \cap C)$. _[We must show that
$x \in A \cap (B \cup C)$.]_
By definition of $\cup$, $x \in A \cap B$ __ (a) __ $x \in A \cap C$.
_Case 1 $(x \in A \cap B)$:_ In this case, by definition of $\cap$, $x \in A$
and $x \in B$$.
Since $x \in B$, then $x \in B \cup C$ by definition of $\cup$.
_Case 2 $(x \in A \cap C)$:_ In this case, by definition of $\cap$, $x \in A$ __
(b) __ $x \in C$.
Since $x \in C$, then $x \in B \cup C$ by definition of $\cup$.
In both cases $x \in A$ and $$ix \in B \cup C, and so, by definition of $\cap$,
__ \(c\) __.
_[So $(A \cap B) \cup (A \cap C) \subseteq A \cap (B \cup C)$ by definition of
__ (d) __ .]_
(3) Conclusion: _[Since both subset relations have been proved, it follows, by
definition of set equality, that __ (a) __.]_
Use an element argument to prove each statement in 7-22. Assume that all sets
are subsets of a universal set $U$.
7. For all sets $A$ and $B$, $(A \cap B)^c = A^c \cup B^c$.
8. For all sets $A$ and $B$, $(A \cap B) \cup (A \cap B^c) = A$.
(This property is used in Section 9.9.)
9. For all sets $A$, $B$, and $C$,
$$ (A - B) \cup (C - B) = (A \cup C) - B $$
10. For all sets $A$, $B$, and $C$,
$$ (A \cup B) \cap C \subseteq A \cup (B \cap C) $$
11. For all sets $A$, $B$, and $C$,
$$ A \cap (B - C) \subseteq (A \cap B) - (A \cap C) $$
12. For all sets $A$, $B$, and $C$,
$$ (A \cup B) - C \subseteq (A - C) \cup (B - C) $$
13. For all sets $A$, $B$, and $C$,
$$ (A - B) \cap (C - B) = (A \cap C) - B $$
14. For all sets $A$ and $B$, $A \cup (A \cap B) = A$.
15. For every set $A$, $A \cup \emptyset = A$.
16. For all sets $A$, $B$, and $C$, if $A \subseteq B$ then
$A \cap C \subseteq B \cap C$.
17. For all sets $A$, $B$, and $C$, if $A \subseteq B$ then
$A \cup C \subseteq B \cup C$.
18. For all sets $A$ and $B$, if $A \subseteq B$ then $B^c \subseteq A^c$.
19. For all sets $A$, $B$, and $C$, if $A \subseteq B$ and $A \subseteq C$ then
$A \subseteq B \cap C$.
20. For all sets $A$, $B$, and $C$, if $A \subseteq C$ and $B \subseteq C$ then
$A \cup B \subseteq C$.
21. For all sets $A$, $B$, and $C$,
$$ A \times (B \cup C) = (A \times B) \cup (A \times C) $$
22. For all sets $A$, $B$, and $C$,
$$ A \times (B \cap C) = (A \times B) \cap (A \times C) $$
23. Find the mistake in the following "proof" that for all sets $A$, $B$, and
$C$, if $A \subseteq B$ and $B \subseteq C$ then $A \subseteq C$.
**Proof:** Suppose $A$, $B$, and $C$ are any sets such that $A \subseteq B$ and
$B \subseteq C$. Since $A \subseteq B$, there is an element $x$ such that
$x \in A$ and $x \in B$, and since $B \subseteq C$, there is an element $x$ such
that $x \in B$ and $x \in C$. Hence there is an element $x$ such that $x \in A$
and $x \in C$ and so $A \subseteq C$.
24. Find the mistake in the following "proof."
**Theorem:** For all sets $A$ and $B$, $A^c \cup B^c \subseteq (A \cup B)^c^c$
**Proof:** Suppose $A$ and $B$ are any sets, and $x \in A^c \cup B^c$. Then
$x \in A^c$ or $x \in B^c$ by definition of union. It follows that $x \notin A$
or $x \notin B$ by definition of complement, and so $x \notin A \cup B$ by
definition of union. Thus $x \in (A \cup B)^c$ by definition of complement, and
hence $A^c \cup B^c \subseteq (A \cup B)^c$.
25. Find the mistake in the following "proof" that for all sets $A$ and $B$,
$(A - B) \cup (A \cap B) \subseteq A$.
**Proof:** Suppose $A$ and $B$ are any sets, and suppose
$x \in (A - B) \cup (A \cap B)$. If $x \in A$ then $x \in A - B$, and so, by
definition of difference, $x \in A$ and $x \notin B$. In particular, $x \in A$,
and, therefore, $(A - B) \cup (A \cap B) \subseteq A$ by definition of subset.
26. Consider the Venn diagram below.
(See page 429 for image.)
a. Illustrate one of the distributive laws by shading in the region
corresponding to $A \cup (B \cap C)$ on one copy of the diagram and
$(A \cup B) \cap (A \cup C)$ on another.
b. Illustrate the other distributive law by shading in the region corresponding
to $A \cap (B \cup C)$ on one copy of the diagram and
$(A \cap B) \cup (A \cap C)$ on another.
c. Illustrate one of De Morgan's laws by shading in the region corresponding to
$(A \cup B)^c$ on one copy of the diagram and $A^c \cap B^c$ on the other.
(Leave the set $C$ out of your diagrams.)
d. Illustrate the other De Morgan's law by shading in the region corresponding
to $(A \cap B)^c$ on one copy of the diagram and $A^c \cup B^c$ on the other.
(Leave the set $C$ out of your diagrams.)
27. Fill in the blanks in the following proof that for all sets $A$ and $B$,
$(A - B) \cap (B - A) = \emptyset$.
**Proof:**
Let $A$ and $B$ be any sets and suppose $(A - B) \cap (B - A) \neq \emptyset$.
That is, suppose there is an element $x$ in __ (a) __. BY definition of __ (b)
__, $x \in A - B$ and $x \in$ __ \(c\) __. Then by definition of set difference,
$x \in A$ and $x \notin B$ and $x \in$ __ (d) __ and $x \notin$ __ (e) __. IN
particular $x \in A$ and $x \notin$ __ (f) __, which is a contradiction. Hence
_[the supposition that $(A - B) \cap (B - A) \neq \emptyset$ is false, and so]_
__ (g) __.
Use the element method for proving a set equals the empty set to prove each
statement in 28-38. Assume that all sets are subsets of a universal set $U$.
28. For all sets $A$ and $B$, $(A \cap B) \cap (A \cap B^c) = \emptyset$. (This
property is used in Section 9.9.)
29. For all sets $A$, $B$, and $C$,
$$ (A - C) \cap (B - C) \cap (A - B) = \emptyset $$
30. For every subset $A$ of a universal set $U$, $A \cap A^c = \emptyset$.
31. If $U$ denotes a universal set, then $U^c = \emptyset$.
32. For every set $A$, $A \times \emptyset = \emptyset$.
33. For all sets $A$ and $B$, if $A \subseteq B$ then $A \cap B^c = \emptyset$.
34. For all sets $A$ and $B$, if $B \subseteq A^c$ then $A \cap B = \emptyset$.
35. For all sets $A$, $B$, and $C$, if $A \subseteq B$ and
$B \cap C = \emptyset$ then $A \cap C = \emptyset$.
36. For all sets $A$, $B$, and $C$, if $C \subseteq B - A$, then
$A \cap C = \emptyset$.
37. For all sets $A$, $B$, and $C$, if $B \cap C \subseteq A$, then
$(C - A) \cap (B - A) = \emptyset$.
38. For all sets $A$, $B$, $C$, and $D$, if $A \cap C = \emptyset$ then
$(A \times B) \cap (C \times D) = \emptyset$.
Prove each statement in 39-44.
39. For all sets $A$ and $B$,
a. $(A - B) \cup (B - A) \cup (A \cap B) = A \cup B$
b. The sets $(A - B)$, $(B - A)$, and $(A \cap B)$ are mutually disjoint.
40. For every positive integer $n$, if $A$ and $B_1, B_2, B_3, \dots$ are any
sets, then
$$ A \cap \left(\bigcup_{i = 1}^{n}B_i\right) = \bigcup_{i = 1}^{n}(A \cap B_i) $$
41. For every positive integer $n$, if $A_1, A_2, A_3, \dots$ and $B$ are any
sets, then
$$ \bigcap_{i = 1}^{n}(A_i - B) = \left(\bigcup_{i = 1}^{n}A_i\right) - B $$
42. For every positive integer $n$, if $A_1, A_2, A_3, \dots$ and $B$ are any
sets, then
$$ \bigcap_{i = 1}^{n}(A_i - B) = \left(\bigcap_{i = 1}^{n}A_i\right) - B $$
43. For every positive integer $n$, if $A$ and $B_1, B_2, B_3, \dots$ are any
sets, then
$$ \bigcup_{i = 1}^{n}(A \times B_i) = A \times \left(\bigcup_{i = 1}^{n}B_i\right) $$
44. For every positive integer $n$, if $A$ and $B_1, B_2, B_3, \dots$ are any
sets, then
$$ \bigcap_{i = 1}^{n}(A \times B_i) = A \times \left(\bigcap_{i = 1}^{n}B_i\right) $$