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:

Lesson-2_-Cartesian-products-and-subsetsDownload

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?)