Mathematical Reasoning: Writing and Proof (PreTeXt Edition)

Subsection Cartesian Products

When working with Cartesian products, it is important to remember that the Cartesian product of two sets is itself a set. As a set, it consists of a collection of elements. In this case, the elements of a Cartesian product are ordered pairs. We should think of an ordered pair as a single object that consists of two other objects in a specified order. For example,

• If \(a \ne 1\text{,}\) then the ordered pair \(\left( {1,a} \right)\) is not equal to the ordered pair \(\left( {a,1} \right)\text{.}\) That is, \(\left( {1,a} \right) \ne \left( {a,1} \right)\text{.}\)

• If \(A = \left\{ {1,2,3} \right\}\) and \(B = \left\{ {a,b} \right\}\text{,}\) then the ordered pair \(\left( {3,a} \right)\) is an element of the set \(A \times B\text{.}\) That is, \(\left( {3,a} \right) \in A \times B\text{.}\)

• If \(A = \left\{ {1,2,3} \right\}\) and \(B = \left\{ {a,b} \right\}\text{,}\) then the ordered pair \(\left( {5,a} \right)\) is not an element of the set \(A \times B\) since \(5 \notin A\text{.}\) That is, \(\left( {5,a} \right) \notin A \times B\text{.}\)

In Section 5.3, we studied certain properties of set union, set intersection, and set complements, which we called the algebra of sets. We will now begin something similar for Cartesian products. We begin by examining some specific examples in Progress Check 5.30 and a little later in Progress Check 5.32.

Subsection The Cartesian Plane

In Beginning Activity 1, we sketched the graph of the equation \(2x + 3y = 12\) in the \(xy\)-plane. This \(xy\)-plane, with which you are familiar, is a representation of the set \(\mathbb{R} \times \mathbb{R}\) or \(\mathbb{R}^2\text{.}\) This plane is called the Cartesian plane.

The basic idea is that each ordered pair of real numbers corresponds to a point in the plane, and each point in the plane corresponds to an ordered pair of real numbers. This geometric representation of \(\mathbb{R}^2\) is an extension of the geometric representation of \(\mathbb{R}\) as a straight line whose points correspond to real numbers.

Since the Cartesian product \(\mathbb{R}^2\) corresponds to the Cartesian plane, the Cartesian product of two subsets of \(\mathbb{R}\) corresponds to a subset of the Cartesian plane. For example, if \(A\) is the interval \(\left[ {1,3} \right]\text{,}\) and \(B\) is the interval \(\left[ {2,5} \right]\text{,}\) then

A graph of the set \(A \times B\) can then be drawn in the Cartesian plane as shown in Figure 5.31.

This illustrates that the graph of a Cartesian product of two intervals of finite length in \(\mathbb{R}\) corresponds to the interior of a rectangle and possibly some or all of its boundary. The solid line for the boundary in Figure 5.31 indicates that the boundary is included. In this case, the Cartesian product contained all of the boundary of the rectangle. When the graph does not contain a portion of the boundary, we usually draw that portion of the boundary with a dotted line.

Note: A Caution about Notation. The standard notation for an open interval in \(\mathbb{R}\) is the same as the notation for an ordered pair, which is an element of \mbox{\(\mathbb{R} \times \mathbb{R}\)} . We need to use the context in which the notation is used to determine which interpretation is intended. For example,

• If we write \(\left( {\sqrt 2 ,7} \right) \in \mathbb{R} \times \mathbb{R}\text{,}\) then we are using \(\left( {\sqrt 2 ,7} \right)\) to represent an ordered pair of real numbers.

• If we write \(\left( {1,2} \right) \times \left\{ 4 \right\}\text{,}\) then we are interpreting \(\left( {1,2} \right)\) as an open interval. We could write

  \begin{equation*} \left( {1,2} \right) \times \left\{ 4 \right\} = \left\{ {\left. {\left( {x,4} \right)\,} \right|1 \lt x \lt 2} \right\}\!\text{.} \end{equation*}

The following progress check explores some of the same ideas explored in Progress Check 5.30 except that intervals of real numbers are used for the sets.

One purpose of the work in Progress Check 5.30 and Progress Check 5.32 was to indicate the plausibility of many of the results contained in the next theorem.

We will not prove all these results; rather, we will prove Item 2 of Theorem 5.33 and leave some of the rest to the exercises. In constructing these proofs, we need to keep in mind that Cartesian products are sets, and so we follow many of the same principles to prove set relationships that were introduced in Section 5.2 and Section 5.3.

The other thing to remember is that the elements of a Cartesian product are ordered pairs. So when we start a proof of a result such as Item 2 of Theorem 5.33, the primary goal is to prove that the two sets are equal. We will do this by proving that each one is a subset of the other one. So if we want to prove that \(A \times \left( {B \cup C} \right) \subseteq \left( {A \times B} \right) \cup \left( {A \times C} \right)\text{,}\) we can start by choosing an arbitrary element of \(A \times \left( {B \cup C} \right)\text{.}\) The goal is then to show that this element must be in \(\left( {A \times B} \right) \cup \left( {A \times C} \right)\text{.}\) When we start by choosing an arbitrary element of \(A \times \left( {B \cup C} \right)\text{,}\) we could give that element a name. For example, we could start by letting

We can then use the definition of “ordered pair” to conclude that

In order to prove that \(A \times \left( {B \cup C} \right) \subseteq \left( {A \times B} \right) \cup \left( {A \times C} \right)\text{,}\) we must now show that the ordered pair \(u\) from (1) is in \(\left( {A \times B} \right) \cup \left( {A \times C} \right)\text{.}\) In order to do this, we can use the definition of set union and prove that

Since \(u = (x, y)\text{,}\) we can prove (49) by proving that

If we look at the sentences in (48) and (50), it would seem that we are very close to proving that \(A \times \left( {B \cup C} \right) \subseteq \left( {A \times B} \right) \cup \left( {A \times C} \right)\text{.}\) Following is a proof of Item 2 of Theorem 5.33.

Final Note: The definition of an ordered pair in Beginning Activity 2 may have seemed like a lengthy definition, but in some areas of mathematics, an even more formal and precise definition of “ordered pair” is needed. This definition is explored in Activity 32.