More On Sets

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Reading (~15 mins)

For the warmup, it’s enough to review last time’s reading, with special focus on the topic of set cardinality.

Warmup Problems (~40 mins)

Please complete these problems “by hand,” without typing. Pencil/pen and paper is best practice for the quiz, but using a stylus with a tablet is also fine. Then, scan/photograph or otherwise produce an image of your solutions and upload them to Canvas.

Problem 1

Define the two sets

\[ \begin{aligned} A &= \{x \in \mathbb{Z} \;|\; x > 0\;,\; x^2 \leq 32\} \\ B &= \{x \in \mathbb{Z} \;|\; x > 0 \;,\; \log_2 x < 7\} \end{aligned} \]

Use the element method to write a proof that that \(A \subseteq B\).

Hint: Recall that your proof should begin with the phrase “Let \(x\in A\)” and end with a phrase similar to “Therefore, \(x \in B\).”

Hint: You may find it helpful to recall the following property of logarithms. For any \(a > 0\) and \(b \in \mathbb{R}\), \(\log_2a^b = b\log_2a\).

Problem 2

(DMOI 0.3.24)

Let \(X = \{n \in \mathbb{N}: 10 \leq n < 20\}\). Find examples of sets with the following properties and briefly explain why your examples are correct.

  1. A set \(A \subseteq \mathbb{N}\) with \(\lvert A \rvert = 10\) such that \(X \setminus A = \{10, 12, 14\}\).
  2. A set \(B \in \mathcal{P}(X)\) with \(\lvert B \rvert = 5\).
  3. A set \(C \subseteq \mathcal{P}(X)\) with \(\lvert C \rvert = 5\).
  4. A set \(D \subseteq X\times X\) with \(\lvert D \rvert = 5\).
  5. A set \(E \subseteq X\) such that \(\lvert E \rvert \in E\).

Problem 3

In a standard deck of playing cards, there are a total of 52 cards. Each card has a number and a suit.

  • The card numbers can be from 1 to 13.
    • The 1 is usually called the ace. Cards 11-13 are usually called the jack, queen, and king, and are collectively referred to as “face cards.”
  • The card suits are spades, clubs, hearts and diamonds. Spades and clubs are “black” suits while hearts and diamonds are “red” suits.

Part A

Describe two sets \(A\) and \(B\) such that the set \(D = A\times B\) describes a deck of cards. In particular, \(D\) should have 52 elements and you should describe a recipe for figuring out which card corresponds to which element of \(D\).

Part B

Compute the following cardinalities:

  1. The number of cards that have a red suit.
  2. The number of face cards.
  3. The number of cards that have a red suit or which are face cards.

Comment on your answer in item 3. How does it relate to the sum of the answers in items 1 and 2? How do you explain the difference?



© Phil Chodrow, 2023