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Section 1.3 Cartesian Products and Power Sets

Subsection 1.3.1 Cartesian Products

Definition 1.3.1. 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:

\begin{equation*} A\times B = \{(a, b) \mid a \in A \quad\textrm{and}\quad b \in B\}\text{,} \end{equation*}

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

    \begin{equation*} A \times B = \{(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5)\}\text{.} \end{equation*}

    Note that \(|A \times B| = 6 = \lvert A \rvert \times \lvert B \rvert \text{.}\)

  • \begin{equation*} A \times A = \{(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)\}\text{.} \end{equation*}

    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. This will be the set of all ordered \(n\)-tuples where the first component comes from the first set, the second component comes from the second set, ... and the \(n^{th}\) component comes from the \(n^{th}\) set. For example,

\begin{equation*} A \times B \times C = \{(a, b, c):a \in A, b \in B, c \in C\}\text{.} \end{equation*}

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*}

Subsection 1.3.2 Power Sets

Definition 1.3.3. 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 \(\mathcal{P}(A)\text{.}\)

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

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

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

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.

Subsection 1.3.3 SageMath Note: Cartesian Products and Power Sets

The second and fourth cells depend on the cells preceeding them, please evaluate from top to bottom.

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.

Exercises 1.3.4 EXERCISES 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. \(A \times \overline{A}\)

  6. \(B^2\)

  7. \(B^3\)

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

Answer
  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 \(B = \{1, 2, 3, 4, 5, 6\}\text{.}\)

  1. What is \(|A \times B|\text{?}\)

  2. How could you interpret the set \(A \times B\) ?

3.

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

Answer

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

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 \(\lvert A \rvert =n\) ?

Answer

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?

Answer

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

  2. \(16 \textrm{ and } 512\)

8.

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

  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?

Answer

They are equal when \(A=B\text{.}\)

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