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:**

Contents

## Vocabulary

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

## Cartesian Products

#### Definitions and Theorems

- An
**ordered pair**is a list $(x,y)$ of two things, $x$ and $y$, enclosed in parentheses and separated by a comma.- NOTE: unlike a set, the order of the elements is important: $(2,4)$ is NOT the same as $(4,2)$

- The
**Cartesian product**of two sets $A$ and $B$ is another set, written $A \times B,$ and defined as $A \times B=\{(a, b): a \in A, b \in B\}$ **Theorem**. If $A$ and $B$ are finite sets, $|A \times B|=|A| \times|B|$.- An
**ordered triple**is a list $(x,y,z)$. - A
**Cartesian power**, like $\mathbb{R}^{2},$ is simply shorthand for the product of a set with itself $\mathbb{R}^{2}=\mathbb{R} \times \mathbb{R}$ (similar for higher powers: $N^{3}=N \times N \times N$).

#### Examples: Cartesian Products

**Example 1**: If $A=\{p, q, r\}$ and $B=\{w, x\},$ find $A \times B$

**Example 2**: i) Describe the Cartesian product $\mathbb{R} \times \mathbb{R}$.

ii) If $A$ is the closed interval $[0,1]$ and $B$ is the half-open interval $[2,3),$ draw a sketch of $A \times B$

**Example 3**: If $A=\{3,7\}, B=\{2,4\},$ and $C=\{5,9\},$ then:

i) is $(3,2,9) \in A \times B \times C$ ?

ii) is $(3,5,2) \in A \times B \times C$ ?

*VIDEO: Examples – Cartesian Products*

## Subsets

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

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

**Example 4. **If $A=\{2,3,5\}, B=\{2,3,4,5,6,7,8\}$ and $C=\{1,2,3\}$

i) is $A \subseteq B$ ? Why?

ii) is $A \subseteq C$ ? Why?

iii) is $C \subseteq A ?$ Why?

iv) is $A \subseteq A$ ? Why?

v) is $\varnothing \subseteq A$ ? 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, $A \subseteq A$

**Theorem**: The empty set is a subset of every set: for any set $A, \varnothing \subseteq A$

#### Exit Questions

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

- a) If $A=\{\pi, 5\}$ and $B=\{4,7\},$ then
- i) Find $A \times B$ and $B^{2}$
- ii) is $(\pi, 7) \in A \times B ?$
- iii) is $(4,5) \in B \times A ?$ iv) is $(\pi, \pi) \in A^{2} ?$

- b) If $A=\{\{4,5,6\}, \varnothing\}$ and $B=\{N, Z,(\varnothing,\{2,7\})\},$ then
- i) is $(\{4,5,6\}, Z) \in A \times B ?$
- ii) is $(\varnothing, \varnothing) \in A \times B ?$
- iii) Find $A \times B .$ What is $|A \times B| ?$ iv) is $((\varnothing,\{2,7\}), \mathrm{N},\{4,5,6\}) \in B^{3} ?$ What product of $\mathrm{A}$ ‘s and $\mathrm{B}$’s is it an element of?

- c) Sketch each set in the plane.
- i) $[1,2] \times(3,5)$
- ii) $\left(-1,-\frac{1}{2}\right] \times[2,3)$
- ii) $[0,1] \times{1,3,5}$

- d) Consider the set with two elements $\{5,\{5\}\} .$ True or False:
- i) $5 \in\{5,\{5\}\}$
- ii) $5 \subseteq\{5,\{5\}\}$
- iii) $\{5\}\in\{5,\{5\}\}$
- iv) $\{5\}\subseteq\{5,\{5\}\}$
- v) $\{\{5\}\}\in\{5,\{5\}\}$
- vi) {{5}}$\subseteq{5,{5}}$

- e) True or False:
- i) $\{(1,1),(2,6),(5,-1),(3,2)\} \subseteq Z \times Z$
- ii) $\mathrm{N} \times \mathrm{N} \subseteq \mathrm{R} \times \mathrm{R}$

#### 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 $\subseteq$ sign, and 5 is not a set)*

d.vi) False

e.i) True (*why?)*

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