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
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Figure 1.15 shows Venn diagrams for these sets.
Fig.1.15 - Venn diagrams for different sets.
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Problem
Using Venn diagrams, verify the following identities.
- $A=(A \cap B) \cup (A-B)$
- If $A$ and $B$ are finite sets, we have $$|A \cup B |=|A|+|B|-|A \cap B| \hspace{120pt} (1.2)$$
- Solution
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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.
Fig.1.16 - Venn diagrams for some identities.
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Problem
Let $S=\{1,2,3\}$. Write all the possible partitions of $S$.
- Solution
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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 $S=\{1,2,3\}$:
- $\{1\},\{2\},\{3\}$;
- $\{1,2\},\{3\}$;
- $\{1,3\},\{2\}$;
- $\{2,3\},\{1\}$;
- $\{1,2,3\}$.
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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
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- $A=\{ x \in \mathbb{Q} | -100 \leq x \leq 100 \}$ 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 one-to-one correspondence with the set of natural numbers. In particular, you can list all the elements in the set $D$, $D=\{1, \frac{1}{2}, \frac{1}{3},\cdots\}$.
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Problem
Find the range of the function $f:\mathbb{R} \rightarrow \mathbb{R}$ defined as $f(x)=\textrm{sin} (x)$.
- Solution
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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$(f)=[-1,1]$.
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