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:

1. 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 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)