Skip to main content

Beginning Activity Beginning Activity 2: The Cartesian Product of Two Sets
A4US

In Beginning Activity 1, we worked with ordered pairs without providing a formal definition of an ordered pair. We instead relied on your previous work with ordered pairs, primarily from graphing equations with two variables. Following is a formal definition of an ordered pair.

Definition.

Let \(A\) and \(B\) be sets. An ordered pair (with first element from \(A\) and second element from \(B\)) is a single pair of objects, denoted by \((a, b)\) , with \(a \in A\) and \(b \in B\) and an implied order. This means that for two ordered pairs to be equal, they must contain exactly the same objects in the same order. That is, if \(a, c \in A\) and \(b, d \in B\text{,}\) then

\begin{equation*} \left( {a,b} \right) = \left( {c,d} \right) \text{ if and only if } a = c \text{ and } b = d\text{.} \end{equation*}

The objects in the ordered pair are called the coordinates of the ordered pair. In the ordered pair \(\left( {a,b} \right)\text{,}\) \(a\) is the first coordinate and \(b\) is the second coordinate.

We will now introduce a new set operation that gives a way of combining elements from two given sets to form ordered pairs. The basic idea is that we will create a set of ordered pairs.

Definition.

If \(A\) and \(B\) are sets, then the Cartesian product, \(A \times B\text{,}\) of \(A\) and \(B\) is the set of all ordered pairs \(\left( {x,y} \right)\) where \(x \in A\) and \(y \in B\text{.}\) We use the notation \(A \times B\) for the Cartesian product of \(A\) and \(B\text{,}\) and using set builder notation, we can write \(A \times B = \left\{ { {\left( {x,y} \right)} \mid x \in A\text{ and } y \in B} \right\}\text{.}\) We frequently read \(A \times B\) as ā€œ\(A\) cross \(B\text{.}\)ā€ In the case where the two sets are the same, we will write \(A^2\) for \(A \times A\text{.}\) That is,

\begin{equation*} A^2 = A \times A = \left\{ {\left( {a,b} \right) \mid a \in A\text{ and } b \in A} \right\}\text{.} \end{equation*}

Let \(A = \left\{ {1,2,3} \right\}\) and \(B = \left\{ {a,b} \right\}\text{.}\)

1.

Is the ordered pair \(\left( {3,a} \right)\) in the Cartesian product \(A \times B\text{?}\) Explain.

2.

Is the ordered pair \(\left( {3,a} \right)\) in the Cartesian product \(A \times A\text{?}\) Explain.

3.

Is the ordered pair \(\left( {3,1} \right)\) in the Cartesian product \(A \times A\text{?}\) Explain.

4.

Use the roster method to specify all the elements of \(A \times B\text{.}\) (Remember that the elements of \(A \times B\) will be ordered pairs.)

5.

Use the roster method to specify all of the elements of the set \(A \times A = A^2\text{.}\)

6.

For any sets \(C\) and \(D\text{,}\) explain carefully what it means to say that the ordered pair \(\left( {x,y} \right)\) is not in the Cartesian product \(C \times D\text{.}\)