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Chapter 1 Sets

Exercises Practice Problems

1. Element, Subset or Neither?

Mark each of the following statements as true or false

  1. Let \(A = \{ 1, \{ 1 \} \}\text{.}\)

    1. \(\displaystyle 1 \in A\)

    2. \(\displaystyle \{ 1 \} \in A\)

    3. \(\displaystyle \{ \{ 1 \} \} \in A\)

  2. Let \(B = \{ 1, 2, 3 \}\)

    1. \(\displaystyle 3 \in B\)

    2. \(\displaystyle \{ 3 \} \in B\)

    3. \(\displaystyle \{ 3 \} \subseteq B\)

  3. \(C = \{ \{1,2\}, \{ \emptyset \} \}\)

    1. \(\displaystyle \emptyset \in C \)

    2. \(\displaystyle \emptyset \subseteq C \)

    3. \(\displaystyle |C|=3\)

  4. \(D = \{ \{1, 2 \}, \{ 3, 4, 5 \} \}\)

    1. \(\displaystyle |D|=5 \)

    2. \(\displaystyle \{ 1, 2 \} \subseteq D \)

    3. \(\displaystyle 4 \in D\)

  5. \(E = \{ \emptyset, \{ \emptyset \} , 0 \}\)

    1. \(\displaystyle \emptyset \in E\)

    2. \(\displaystyle \{ \{ \emptyset \} \} \subseteq E \)

    3. \(\displaystyle |E|=3\)

2. Counting Subsets.

Let \(A\) be any set. In this problem, we will see why mathematicians decided that is it "right" to say that \(\emptyset \subseteq A\) and \(A \subseteq A\text{.}\)

  1. List all of the subsets of \(\{ 1 \}\text{.}\)

  2. List all of the subsets of \(\{ 1,2 \}\text{.}\)

  3. List all of the subsets of \(\{ 1,2,3 \}\text{.}\)

  4. Look at the pattern you see so far. Guess how many subsets are there of the set \(\{ 1,2,3,4 \}\text{.}\) How about the set \(\{ 1,2,3, \ldots , 10 \}\text{?}\)

  5. Your answers in part (d) are very simple expressions. This is one reason why mathematicians defined "subset" to include the cases \(\emptyset \subseteq A\) and \(A \subseteq A\text{.}\) The power set of a set \(A\) is the set of all subsets of \(A\text{.}\) This power set is denoted \(\mathcal{P}(A)\text{.}\) If \(|A|=n\text{,}\) what is \(| \mathcal{P}(A)|\text{?}\) (You do not have to prove your answer.)

3. Empty Sets All the Way Down.

Consider the following sequence of sets \[ \emptyset, \{ \emptyset \}, \Big\{ \emptyset, \{ \emptyset \} \Big\}, \bigg\{ \emptyset , \{ \emptyset \} , \Big\{ \emptyset, \{ \emptyset \} \Big\} \bigg\}. \]

  1. What is the size of each of these sets?

  2. What is the next set in this series?

  3. Dr. Philomath says, "You can create the natural numbers from nothing." What does she mean by this statement? Do you think that this is a true statement or a false statement?

4. The Barber Paradox.

Consider the following classic paradox: "The barber is a man in town who shaves all those (and only those) men in town who do not shave themselves. Who shaves the barber?"

Note that I've updated the classic phrasing so that it is actually a paradox. Also, let us assume that every man shaves (even if it's just to clean up his beard).

  1. Explain the logical paradox of this description of the barber.

  2. The classic formulation of the "barber paradox" is actually: "The barber shaves every man in town who does not shave themselves. Who shaves the barber?" There are multiple reasons why this statement is NOT a paradox. Can you find at least one?

5. A Weird Set.

Let \(S\) be the set of sets that do not contain themselves. Consider the following two statements: \[ S \in S \mbox{ and } S \subseteq S. \] Which of these statements is true, and which of these statements is a paradox?