# 1.3: Cartesian Products and Power Sets

- Page ID
- 80497

## Cartesian Products

Definition \(\PageIndex{1}\): Cartesian 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 \(B\text{.}\)

Example \(\PageIndex{1}\): Cartesian Product

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:

and in general,

## Power Sets

Definition \(\PageIndex{2}\): Power 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 \(\mathcal{P}(A)\text{.}\)

Example \(\PageIndex{2}\): Some Power Sets

- \(\displaystyle \mathcal{P}(\emptyset )=\{\emptyset \}\)
- \(\displaystyle \mathcal{P}(\{1\}) = \{\emptyset , \{1\}\}\)
- \(\mathcal{P}(\{1,2\}) = \{\emptyset , \{1\}, \{2\}, \{1, 2\}\}\text{.}\)

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.

## SageMath Note: Cartesian Products and Power Sets

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

A=Set([0,1,2]) B=Set(['a','b']) P=cartesian_product([A,B]);P

Here is the cardinality of the cartesian product.

P.cardinality()

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

U=Set([0,1,2,3]) subsets(U)

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

for a in subsets(U): print(str(a)+ " has " +str(len(a))+" elements.")

### Exercises

Exercise \(\PageIndex{1}\)

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

- \(\displaystyle A \times B\)
- \(\displaystyle B \times A\)
- \(\displaystyle A \times B\times C\)
- \(\displaystyle U \times \emptyset\)
- \(\displaystyle A \times A^c\)
- \(\displaystyle B^2\)
- \(\displaystyle B^3\)
- \(\displaystyle B\times \mathcal{P}(B)\)

**Answer**-
- \(\displaystyle \{(0, 2), (0, 3), (2, 2), (2, 3), (3, 2), (3, 3)\}\)
- \(\displaystyle \{(2, 0), (2, 2), (2, 3), (3, 0), (3, 2), (3, 3)\}\)
- \(\displaystyle \{(0, 2, 1), (0, 2, 4), (0, 3, 1), (0, 3, 4), (2, 2, 1), (2, 2, 4),\\ (2, 3, 1), (2, 3, 4), (3, 2, 1), (3, 2, 4), (3, 3, 1), (3, 3, 4)\}\)
- \(\displaystyle \emptyset\)
- \(\displaystyle \{(0, 1), (0, 4), (2, 1), (2, 4), (3, 1), (3, 4)\}\)
- \(\displaystyle \{(2, 2), (2, 3), (3, 2), (3, 3)\}\)
- \(\displaystyle \{(2, 2, 2), (2, 2, 3), (2, 3, 2), (2, 3, 3), (3, 2, 2), (3, 2, 3), (3, 3, 2), (3, 3, 3)\}\)
- \(\displaystyle \{(2, \emptyset ), (2, \{2\}), (2, \{3\}), (2, \{2, 3\}), (3, \emptyset ), (3, \{2\}), (3, \{3\}), (3, \{2, 3\})\}\)

Exercise \(\PageIndex{2}\)

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 \(A \times B\) ?

Exercise \(\PageIndex{3}\)

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

**Answer**-
\(\{a, b\}, \{a, c\}, \{a, d\}, \{b, c\}, \{b, d\} \textrm{ and } \{c, d\}\)

Exercise \(\PageIndex{4}\)

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

Exercise \(\PageIndex{5}\)

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

**Answer**-
There are \(n\) singleton subsets, one for each element.

Exercise \(\PageIndex{6}\)

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?

Exercise \(\PageIndex{7}\)

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?

**Answer**-
- \(\displaystyle \{+00, +01, +10, +11, -00, -01, -10, -11\}\)
- \(\displaystyle 16 \textrm{ and } 512\)

Exercise \(\PageIndex{8}\)

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.

Exercise \(\PageIndex{9}\)

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

**Answer**-
They are equal when \(A=B\).