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:


  • 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


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?)

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?)