
Section1.3Cartesian Products and Power Sets

Subsection1.3.1Cartesian Products

Definition1.3.1Cartesian Product

Let $A$ and $B$ be sets. The Cartesian product of $A$ and $B\text{,}$ denoted by $A\times B\text{,}$ is defined as follows: $A\times B = \{(a, b) \mid a \in A \quad\textrm{ and }\quad b \in B\}\text{,}$ that is, $A\times B$ is the set of all possible ordered pairs whose first component comes from $A$ and whose second component comes from $B\text{.}$

Notation in mathematics is often developed for good reason. In this case, a few examples will make clear why the symbol $\times$ is used for Cartesian products.

• Let $A = \{1, 2, 3\}$ and $B = \{4, 5\}\text{.}$ Then $A \times B = \{(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5)\}\text{.}$ Note that $|A \times B| = 6 = \lvert A \rvert \times \lvert B \rvert \text{.}$

• $A \times A = \{(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)\}\text{.}$ Note that $|A \times A| = 9 = {\lvert A \rvert}^2\text{.}$

These two examples illustrate the general rule that if $A$ and $B$ are finite sets, then $\lvert A \times B \rvert = \lvert A \rvert \times \lvert B \rvert \text{.}$

We can define the Cartesian product of three (or more) sets similarly. For example, $A \times B \times C = \{(a, b, c):a \in A, b \in B, c \in C\}\text{.}$

It is common to use exponents if the sets in a Cartesian product are the same:

\begin{equation*} A^2= A \times A \end{equation*}
\begin{equation*} A^3=A \times A \times A \end{equation*}

and in general,

\begin{equation*} A^n = \underset{n \textrm{ factors}}{\underline{A \times A \times \ldots \times A}}\text{.} \end{equation*}

Subsection1.3.2Power Sets

Definition1.3.3Power Set

If $A$ is any set, the power set of $A$ is the set of all subsets of $A\text{,}$ denoted $\mathcal{P}(A)\text{.}$

The two extreme cases, the empty set and all of $A\text{,}$ are both included in $\mathcal{P}(A)\text{.}$

• $\mathcal{P}(\emptyset )=\{\emptyset \}$

• $\mathcal{P}(\{1\}) = \{\emptyset , \{1\}\}$

• $\mathcal{P}(\{1,2\}) = \{\emptyset , \{1\}, \{2\}, \{1, 2\}\}\text{.}$

We will leave it to you to guess at a general formula for the number of elements in the power set of a finite set. In Chapter 2, we will discuss counting rules that will help us derive this formula.

Subsection1.3.3SageMath Note: Cartesian Products and Power Sets

Here is a simple example of a cartesian product of two sets:

Here is the cardinality of the cartesian product.

The power set of a set is an iterable, as you can see from the output of this next cell

You can iterate over a powerset. Here is a trivial example.

Subsection1.3.4EXERCISES FOR SECTION 1.3

1

Let $A = \{0, 2, 3\}\text{,}$ $B = \{2, 3\}\text{,}$ $C = \{1, 4\}\text{,}$ and let the universal set be $U = \{0, 1, 2, 3, 4\}\text{.}$ List the elements of

1. $A \times B$

2. $B \times A$

3. $A \times B\times C$

4. $U \times \emptyset$

5. $A \times A^c$

6. $B^2$

7. $B^3$

8. $B\times \mathcal{P}(B)$

1. $\{(0, 2), (0, 3), (2, 2), (2, 3), (3, 2), (3, 3)\}$
2. $\{(2, 0), (2, 2), (2, 3), (3, 0), (3, 2), (3, 3)\}$
3. $\{(0, 2, 1), (0, 2, 4), (0, 3, 1), (0, 3, 4), (2, 2, 1), (2, 2, 4),\\ (2, 3, 1), (2, 3, 4), (3, 2, 1), (3, 2, 4), (3, 3, 1), (3, 3, 4)\}$
4. $\emptyset$
5. $\{(0, 1), (0, 4), (2, 1), (2, 4), (3, 1), (3, 4)\}$
6. $\{(2, 2), (2, 3), (3, 2), (3, 3)\}$
7. $\{(2, 2, 2), (2, 2, 3), (2, 3, 2), (2, 3, 3), (3, 2, 2), (3, 2, 3), (3, 3, 2), (3, 3, 3)\}$
8. $\{(2, \emptyset ), (2, \{2\}), (2, \{3\}), (2, \{2, 3\}), (3, \emptyset ), (3, \{2\}), (3, \{3\}), (3, \{2, 3\})\}$
2

Suppose that you are about to flip a coin and then roll a die. Let $A = \{HEADS, TAILS\}$ and $B = \{1, 2, 3, 4, 5, 6\}\text{.}$

• What is $|A \times B|\text{?}$

• How could you interpret the set $A \times B$ ?

3

List all two-element sets in $\mathcal{P}(\{a,b,c,d\})$

$\{a, b\}, \{a, c\}, \{a, d\}, \{b, c\}, \{b, d\} \textrm{ and } \{c, d\}$

4

List all three-element sets in $\mathcal{P}(\{a, b, c,d\})\text{.}$

5

How many singleton (one-element) sets are there in $\mathcal{P}(A)$ if $\lvert A \rvert =n$ ?

There are $n$ singleton subsets, one for each element.

6

A person has four coins in his pocket: a penny, a nickel, a dime, and a quarter. How many different sums of money can he take out if he removes 3 coins at a time?

7

Let $A = \{+,-\}$ and $B = \{00, 01, 10, 11\}\text{.}$

• List the elements of $A \times B$

• How many elements do $A ^4$ and $(A \times B)^3$ have?

1. $\{+00, +01, +10, +11, -00, -01, -10, -11\}$

2. $16 \textrm{ and } 512$

8

Let $A = \{\bullet,\square ,\otimes \}$ and $B = \{\square ,\ominus ,\bullet\}\text{.}$

• List the elements of $A \times B$ and $B \times A\text{.}$ The parentheses and comma in an ordered pair are not necessary in cases such as this where the elements of each set are individual symbols.

• Identify the intersection of $A \times B$ and $B \times A$ for the case above, and then guess at a general rule for the intersection of $A \times B$ and $B \times A\text{,}$ where $A$ and $B$ are any two sets.

9

Let $A$ and $B$ be nonempty sets. When are $A \times B$ and $B \times A$ equal?

They are equal when $A=B$.