More On Sets

Videos (15 mins)

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

(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 \(\lvert E \rvert \in E\).

Hint: Recall that \(\mathcal{P}(A)\) is the power set of \(A\), the set of all subsets of \(A\).

Problem 2

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, 2024