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Exercises 1.4 Exercises

1.

Write the set of prime numbers less than \(10\) in roster notation.

2.

Write the set of letters in the alphabet in roster notation. You can use ellipses (\(\ldots\)) once the pattern is clear.

Solution.

Solution: \(\{a, b, c, \ldots, z\}\)

3.

Write the set of positive odd integers in roster notation, using ellipses (\(\ldots\)) once the pattern is clear.

4.

Write the set \(\{a,e,i,o,u\}\) in set-builder notation.

5.

Write the set \(\{2,4,6,8,10,\ldots\}\) in set-builder notation.

6.

Write the set \(\Big\{ \varnothing, \{a\}, \{b\}, \{a,b\} \Big\}\) in set-builder notation.

Hint.

Hint: Hint: What do these sets all have in common?

Solution.

Solution: These elements are all the subsets of \(\{a,b\}\text{.}\) So, this set can be written as

\begin{equation*} \Big\{ E \; | \; E \subseteq \{a,b\} \Big\} \end{equation*}
7.

Calculate the power set of \(X=\{1,2,3,4\}\text{.}\)

8.

Calculate the power set of \(D=\{N,S,W,E\}\text{.}\)

9.

Calculate the power set of \(Y=\{a,b,c,d,e\}\text{.}\)

Solution.

Solution: When listing large collections it is useful to construct a program to follow. It turns out there is one empty subset, five singleton subsets, ten pairs, ten triples, five sets of four, and one set of five. (How do you know? Wait for Part III...)

\begin{align*} \mathscr{P}(\{a,b,c,d,e\}) = \Big\{ \amp \varnothing, \{a\}, \{b\}, \{c\}, \{d\}, \{e\},\\ \amp \{a,b\}, \{a,c\}, \{a,d\}, \{a,e\}, \{b,c\},\\ \amp \{b,d\}, \{b,e\}, \{c,d\}, \{c,e\}, \{d,e\},\\ \amp \{a,b,c\}, \{a,b,d\}, \{a,b,e\}, \{a,c,d\}, \{a,c,e\},\\ \amp \{a,d,e\}, \{b,c,d\}, \{b,d,e\}, \{b,c,e\}, \{c,d,e\},\\ \amp \{a,b,c,d\}, \{a,b,c,e\}, \{a,b,d,e\},\\ \amp \{a,c,d,e\}, \{b,c,d,e\}, \{a,b,c,d,e\}\Big\} \end{align*}
10.

Calculate \(\mathscr{P}\Big(\mathscr{P}(\{a\})\Big)\text{.}\)

Solution.

Solution: The power set of \(\{a\}\) is

\begin{equation*} \mathscr{P}(\{a\}) = \Big\{\varnothing, \{a\}\Big\} \end{equation*}

The power set of \(\mathscr{P}(\{a\})\) is

\begin{equation*} \mathscr{P}\Big(\mathscr{P}(\{a\})\Big) = \Big\{\varnothing, \{\varnothing\}, \{\{a\}\}, \{\varnothing, \{a\}\}\Big\} \end{equation*}
11.

Calculate the power set of the power set of \(\{a,b\}\text{.}\)

12.

Calculate the power set of the power set of the power set of \(\varnothing\text{.}\)

Hint.

Hint: This collection contains four elements.

13.

Tell whether each statement is true or false if \(A=\{a,b\}\text{.}\)

  1. \(\displaystyle a \in \mathscr{P}(A)\)

  2. \(\displaystyle \{a\} \in \mathscr{P}(A)\)

  3. \(\displaystyle \{\{a\}\} \in \mathscr{P}(A)\)

  4. \(\displaystyle \{\{a\}\} \subseteq \mathscr{P}(A)\)

14.

Which if any of the following objects are an element of \(\mathscr{P}(\mathbb{N})\text{:}\) \(3\text{,}\) \(\{3\}\text{,}\) \(\{\{3\}\}\text{,}\) \(\varnothing\text{,}\) and/or \(\{\varnothing\}\text{?}\)

15.

Which if any of the following objects are an element of \(\mathscr{P}(\mathbb{Z})\text{:}\) \(-2.3\text{,}\) \(\{-2.3\}\text{,}\) \(\{\{-2.3\}\}\text{,}\) \(\varnothing\text{,}\) and/or \(\{\varnothing\}\text{?}\)

16.

In one symbol, what is \(\mathbb{N}\cup \mathbb{Z}\text{?}\)

Solution.

Solution: Because every natural number is an integer anyway, \(\mathbb{N}\cup \mathbb{Z}\) is just \(\mathbb{Z}\text{.}\)

17.

In one symbol, what is \(\mathbb{N}\cap \mathbb{Z}\text{?}\)

18.

In one symbol, what is \(\mathbb{N}-\mathbb{Z}\text{?}\)

19.

Suppose \(A \subseteq B\text{.}\) What are \(A \cup B\text{,}\) \(A \cap B\text{,}\) and \(A - B\text{?}\)

Hint.

Hint: Last three problems.

20.

Given \(A=\{a,b,c\}\) and \(B=\{c,d,e\}\text{,}\) compute the following. Give your answer in list/roster notation.

  1. \(\displaystyle A \cup B\)

  2. \(\displaystyle A \cap B\)

  3. \(\displaystyle A-B\)

  4. \(\displaystyle B \times A\)

21.

Let \(X=\{1,3,5\}\text{,}\) \(Y=\{2,4,6\}\text{,}\) and \(Z=\{2,3,5\}\text{.}\) Consider the universal set to be \(\Omega=\{1,2,3,4,5,6\}\text{.}\) Compute the following. Give your answer in list/roster notation.

  1. \(\displaystyle X \cup Y\)

  2. \(\displaystyle X \cap Z\)

  3. \(\displaystyle \overline{Z}\)

  4. \(\displaystyle X \times Z\)

  5. \(\displaystyle X \cup (Y \cap Z)\)

22.

Let \(E=\{w,x,y,z\}\text{,}\) \(F=\{u,v,w\}\text{,}\) and \(G=\{t,v,x\}\) be contained in the universal set \(\Omega=\{s,t,u,v,w,x,y,z\}\text{.}\) Give your answer in list/roster notation.

  1. \(\displaystyle E \cup F\cup G\)

  2. \(\displaystyle E \cap F\)

  3. \(\displaystyle G - E\)

  4. \(\displaystyle \overline{E}\)

  5. \(\displaystyle G \times F\)

Hint.

Hint: \(E \cup F \cup G\) means \(E \cup (F \cup G)\) (or \((E \cup F) \cup G\)).

23.

Let the universal set \(\Omega=\{0,1,2,3,4,5,6,7,8,9\}\) contain the sets mentioned below. Calculate the following sets. Give your answer in list/roster notation.

  1. \(\displaystyle \{1,2,5\} \cup \{2,5,8,9\}\)

  2. \(\displaystyle \overline{\{2,3,7\}}\)

  3. \(\displaystyle \{x \in \Omega \; | \; x \text{ is odd}\} \cap \{0,3,4,9\}\)

24.

Let the universal set \(\Omega=\{0,1,2,3,4,5,6,7,8,9\}\) contain the sets mentioned below. Calculate the following sets. Give your answer in list/roster notation.

  1. \(\displaystyle \{3,7,9\} \cup \{x \in \Omega \; | \; x \text{ is even}\}\)

  2. \(\displaystyle \{2,5,7,8\} \cap \{1,2,3,4,5\}\)

  3. \(\displaystyle \overline{\{1,4,8,9\}}\)

25.

Let \(X=\{1,3,5\}\text{,}\) \(Y=\{2,4,6\}\text{,}\) and \(Z=\{2,3,5\}\text{.}\) Consider the universal set to be \(\Omega=\{1,2,3,4,5,6\}\text{.}\) Calculate \(|X \cup Y \cup Z|\text{.}\)

26.

Let \(X=\{1,3,5\}\text{,}\) \(Y=\{2,4,6\}\text{,}\) and \(Z=\{2,3,5\}\text{.}\) Consider the universal set to be \(\Omega=\{1,2,3,4,5,6\}\text{.}\) Calculate \(|\overline{Z} \times Y|\text{.}\)

27.

Let \(X=\{1,3,5\}\text{,}\) \(Y=\{2,4,6\}\text{,}\) and \(Z=\{2,3,5\}\text{.}\) Consider the universal set to be \(\Omega=\{1,2,3,4,5,6\}\text{.}\) Calculate \(|\mathscr{P}(Z-Y)|\text{.}\)

28.

Thinking about the formula for the cardinality of a union that you learned, what is a similar formula for \(|A \cup B \cup C|\text{?}\)

Hint.

Hint: It makes sense to start with \(|A|+|B|+|C|\text{.}\) Then what do you need to subtract? Are you done? Try tracking an element \(x \in A \cap B \cap C\) and seeing how many times your formula has counted it.