1.2.5 Solved Problems:
Review of Set Theory
Let $A$, $B$, $C$ be three sets as shown in the following Venn diagram. For each of the following sets, draw a Venn diagram and shade the area representing the given set.
 $A \cup B \cup C$
 $A \cap B \cap C$
 $A \cup (B \cap C)$
 $A(B \cap C)$
 $A \cup (B \cap C)^c$
 Solution

Figure 1.15 shows Venn diagrams for these sets.

Problem
Using Venn diagrams, verify the following identities.
 $A=(A \cap B) \cup (AB)$
 If $A$ and $B$ are finite sets, we have $$A \cup B =A+BA \cap B \hspace{120pt} (1.2)$$
 Solution

Figure 1.16 pictorially verifies the given identities. Note that in the second identity, we show the number of elements in each set by the corresponding shaded area.

Problem
Let $S=\{1,2,3\}$. Write all the possible partitions of $S$.
 Solution

Remember that a partition of $S$ is a collection of nonempty sets that are disjoint and their union is $S$. There are $5$ possible partitions for $S=\{1,2,3\}$:
 $\{1\},\{2\},\{3\}$;
 $\{1,2\},\{3\}$;
 $\{1,3\},\{2\}$;
 $\{2,3\},\{1\}$;
 $\{1,2,3\}$.

Problem
Determine whether each of the following sets is countable or uncountable.
 $A=\{ x \in \mathbb{Q}  100 \leq x \leq 100 \}$
 $B=\{(x,y)  x \in \mathbb{N}, y \in \mathbb{Z} \}$
 $C=(0,0.1]$
 $D=\{ \frac{1}{n}  n \in \mathbb{N} \}$
 Solution

 $A=\{ x \in \mathbb{Q}  100 \leq x \leq 100 \}$ is countable since it is a subset of a countable set, $A \subset \mathbb{Q}$.
 $B=\{(x,y)  x \in \mathbb{N}, y \in \mathbb{Z} \}$ is countable because it is the Cartesian product of two countable sets, i.e., $B= \mathbb{N} \times \mathbb{Z}$.
 $C=(0,.1]$ is uncountable since it is an interval of the form $(a,b]$, where $a < b$.
 $D=\{ \frac{1}{n}  n \in \mathbb{N} \}$ is countable since it is in onetoone correspondence with the set of natural numbers. In particular, you can list all the elements in the set $D$, $D=\{1, \frac{1}{2}, \frac{1}{3},\cdots\}$.

Problem
Find the range of the function $f:\mathbb{R} \rightarrow \mathbb{R}$ defined as $f(x)=\textrm{sin} (x)$.
 Solution

For any real value $x$, $1 \leq \textrm{sin} (x) \leq 1$. Also, all values in $[1,1]$ are covered by $\textrm{sin} (x)$. Thus, Range$(f)=[1,1]$.
