Applied Discrete Structures

Section1.3Cartesian Products and Power Sets

Subsection1.3.1Cartesian Products

Definition1.3.1.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 $$B\text{.}$$
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

Definition1.3.3.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 $$\mathcal{P}(A)\text{.}$$
• $$\displaystyle \mathcal{P}(\emptyset )=\{\emptyset \}$$
• $$\displaystyle \mathcal{P}(\{1\}) = \{\emptyset , \{1\}\}$$
• $$\mathcal{P}(\{1,2\}) = \{\emptyset , \{1\}, \{2\}, \{1, 2\}\}\text{.}$$
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.

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.

Exercises1.3.4Exercises

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. $$\displaystyle A \times B$$
2. $$\displaystyle B \times A$$
3. $$\displaystyle A \times B\times C$$
4. $$\displaystyle U \times \emptyset$$
5. $$\displaystyle A \times A^c$$
6. $$\displaystyle B^2$$
7. $$\displaystyle B^3$$
8. $$\displaystyle B\times \mathcal{P}(B)$$
1. $$\displaystyle \{(0, 2), (0, 3), (2, 2), (2, 3), (3, 2), (3, 3)\}$$
2. $$\displaystyle \{(2, 0), (2, 2), (2, 3), (3, 0), (3, 2), (3, 3)\}$$
3. $$\displaystyle \{(0, 2, 1), (0, 2, 4), (0, 3, 1), (0, 3, 4), (2, 2, 1), (2, 2, 4),\\ (2, 3, 1), (2, 3, 4), (3, 2, 1), (3, 2, 4), (3, 3, 1), (3, 3, 4)\}$$
4. $$\displaystyle \emptyset$$
5. $$\displaystyle \{(0, 1), (0, 4), (2, 1), (2, 4), (3, 1), (3, 4)\}$$
6. $$\displaystyle \{(2, 2), (2, 3), (3, 2), (3, 3)\}$$
7. $$\displaystyle \{(2, 2, 2), (2, 2, 3), (2, 3, 2), (2, 3, 3), (3, 2, 2), (3, 2, 3), (3, 3, 2), (3, 3, 3)\}$$
8. $$\displaystyle \{(2, \emptyset ), (2, \{2\}), (2, \{3\}), (2, \{2, 3\}), (3, \emptyset ), (3, \{2\}), (3, \{3\}), (3, \{2, 3\})\}$$

2.

Suppose that you are about to flip a coin and then roll a die. Let $$A = \{HEADS, TAILS\}$$ and $$B = \{1, 2, 3, 4, 5, 6\}\text{.}$$
1. What is $$|A \times B|\text{?}$$
2. 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 $$B = \{00, 01, 10, 11\}\text{.}$$
1. List the elements of $$A \times B$$
2. How many elements do $$A ^4$$ and $$(A \times B)^3$$ have?
1. $$\displaystyle \{+00, +01, +10, +11, -00, -01, -10, -11\}$$
2. $$\displaystyle 16 \textrm{ and } 512$$

8.

Let $$A = \{\bullet,\square ,\otimes \}$$ and $$B = \{\square ,\ominus ,\bullet\}\text{.}$$
1. 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.
2. 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 $$A=B\text{.}$$