Hi everyone! This online lesson is provided as a resource – we will also go over this material in class.

Lesson 2: Cartesian Products and Subsets

**Topic**. This lesson covers:

- Sec 1.2: Cartesian Products
- Sec 1.3: Subsets

**Learning Outcomes.**

- Identify and manipulate ordered pairs and Cartesian products of sets.
- Identify and manipulate subsets of sets.

**WeBWorK**. There is 1 WeBWorK assignment on today’s material:

- Assignment1-Sec1.2-1.3

**Lecture Notes:**

## Vocabulary

- ordered pair
- Cartesian product
- ordered triple
- Cartesian power
- subset

## Cartesian Products

#### Definitions and Theorems

- An
**ordered pair**is a list of two things, and , enclosed in parentheses and separated by a comma.- NOTE: unlike a set, the order of the elements is important: is NOT the same as

- The
**Cartesian product**of two sets and is another set, written and defined as **Theorem**. If and are finite sets, .- An
**ordered triple**is a list . - A
**Cartesian power**, like is simply shorthand for the product of a set with itself (similar for higher powers: ).

#### Examples: Cartesian Products

**Example 1**: If and find

**Example 2**: i) Describe the Cartesian product .

ii) If is the closed interval and is the half-open interval draw a sketch of

**Example 3**: If and then:

i) is ?

ii) is ?

*VIDEO: Examples – Cartesian Products*

## Subsets

**Definition**. If and are sets and every element of is also an element of , then we say is a **subset** of and we write .

If this is NOT the case then we say is not a subset of and we write . *NOTE: means there is at least one element of A that is not an element of B.*

**Example 4. **If and

i) is ? Why?

ii) is ? Why?

iii) is Why?

iv) is ? Why?

v) is ? Why?

*VIDEO: Example – Subsets*

Take a moment to absorb the following two theorems. Do you believe them? Why or why not?

**Theorem**: Every set is a subset of itself,

**Theorem**: The empty set is a subset of every set: for any set

#### Exit Questions

Test your understanding of products and subsets by working through the following examples (*selected* answers are provided).

- a) If and then
- i) Find and
- ii) is
- iii) is iv) is

- b) If and then
- i) is
- ii) is
- iii) Find What is iv) is What product of ‘s and ‘s is it an element of?

- c) Sketch each set in the plane.
- i)
- ii)
- ii)

- d) Consider the set with two elements True or False:
- i)
- ii)
- iii)
- iv)
- v)
- vi) {{5}}

- e) True or False:
- i)
- ii)

#### Selected Answers

a.ii) Yes

a.iv) Yes

b.i) Yes

b.ii) No *(Why not?)*

c.i)

d.ii) False *(there must be a SET on both sides of a sign, and 5 is not a set)*

d.vi) False

e.i) True (*why?)*

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