Sets and Set Operations

Videos (~40 mins)

Intro to Sets (7:12)
Set Roster vs. Set Builder Notation (5:15)
The Empty Set (4:07)
Cartesian Product (7:10)
Union of Two Sets (5:07)
Intersection of Two Sets (5:47)
Proving Subset Relationships with the Element Method (6:35)

Reading (30 mins)

DMOI 0.3

Optional Reading

This is a slightly more advanced reading that may appeal to you if you want to see more theoretical math examples.

BOP 1.1, 1.4-1.7

Warmup Problems (~30 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

Write the following sets by listing their elements between braces:

\(\left\{ x \in \mathbb{Z} \; : x^2 \leq 10 \right\}\)
\(\left\{ x \in \mathbb{N} \; : x \leq 10 \right\} \cap \left\{ x \in \mathbb{Z} \; : x \text{ is a prime number.} \right\}\)
\(\left\{ x \in \mathbb{Z} \; : x/2 \geq 10 \right\} \cap \left\{ x \in \mathbb{Z} \; : 5x < 100 \right\}\)

Problem 2

Write each of the following sets in set-builder notation. There may be several correct ways.

\(\left\{ 2, 4, 8, 16, 32, 64,\ldots \right\}\)
\(\left\{ 3, 4, 5, 6, 7, 8 \right\}\)
\(\left\{ 0, 3, 6, 9, 12, 15 \right\}\)

Problem 3

Let \(A = \{x \in \mathbb{Z} \;:\; x = 4y+1 \text{ for some } y \in \mathbb{Z}\}\) and let \(B = \{x \in \mathbb{Z} \;:\; x = 2k+1 \text{ for some } k \in \mathbb{Z}\}\).

Part A

Write out at least three distinct elements of \(A\) and three distinct elements of \(B\).

Part B

Is \(7 \in A\)? Is \(7 \in B\)?

Part C

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

Part D

Would your proof still work if we instead defined \(B = \{x \in \mathbb{z} \;:\; x = 2k+1 \text{ for some } k \in \mathbb{N}\}\)? Briefly explain why or why not.