The Cartesian Product

Given two sets A and B, it is possible to “multiply” them to produce a new set denoted as A×B. This operation is called the Cartesian product. To understand it, we must first understand the idea of an ordered pair.

[Definition 1] An ordered pair is a list (x,y) of two things x and y, enclosed in parentheses and separated by a comma.

For example, (2,4) is an ordered pair, as is (4,2). These ordered pairs are different because even though they have the same things in them, the order is different. We write (2,4)≠(4,2). Right away you can see that ordered pairs can be used to describe points on the plane, as was done in calculus, but they are not limited to just that. The things in an ordered pair don’t have to be numbers. You can have ordered pairs of letters, such as (m,ℓ), ordered pairs of sets such as ({2,5},{3,2}), even ordered pairs of ordered pairs like ((2,4),(4,2)). The following are also ordered pairs: (2,{1,2,3}), (ℝ,(0,0)). Any list of two things enclosed by parentheses is an ordered pair. Now we are ready to define the Cartesian product.

[Definition 2] The Cartesian product of two sets A and B is another set, denoted as A×B and defined as A×B={(a,b):a∈A,b∈B}.

Thus A×B is a set of ordered pairs of elements from A and B. For example, if A={k,ℓ,m} and B={q,r}, then

A×B={(k,q),(k,r),(ℓ,q),(ℓ,r),(m,q),(m,r)}.
Figure 1 shows how to make a schematic diagram of A×B. Line up the elements of A horizontally and line up the elements of B vertically, as if A and B form an x-and y-axis. Then fill in the ordered pairs so that each element (x,y) is in the column headed by x and the row headed by y.

 

Figure 1. A diagram of a Cartesian product

For another example, {0,1}×{2,1}={(0,2),(0,1),(1,2),(1,1)}. If you are a visual thinker, you may wish to draw a diagram similar to Figure 1. The rectangular array of such diagrams give us the following general fact.

Fact 1 If A and B are finite sets, then |A×B|=|A|⋅|B|.

Cartesian Product

Definition: Cartesian Product
The Cartesian product of set A and set B, symbolized by A×B and read “A cross B,” is the set of all possible ordered pairs of the form (a,b). where a∈A and b∈B.

To determine the ordered pairs in a Cartesian product, select the first element of set A and form an ordered pair with each element of set B. Then select the second element of set A and form an ordered pair with each element of set B. Continue in this manner until you have used each element of set A.

⛲ Example 1: The Cartesian Product of Two Sets
Given A={orange, banana, apple} and B={1,2},detcrrnine the following.
a) A×B b) B×A c) A×A d) B×B
Solution:
a) A×B={(orange,1),(orange,2) (banana,1),(banana,2),(apple,1),(apple,2)}
b) B×A={(1,orange),(1,banana),(1,apple),(2,orange),(2,banana),(2,apple)}
c) A×A={(orange,orange),(orange,banana),(orange,apple),(banana, orange), (banana, banana), (banana,apple),(apple,orange),(apple,banana),(apple,apple)}
d) B×B={(1,1),(1,2),(2,1),(2,2)}

We can see from Example 1 that. in general, A×B≠B×A. The ordered pairs in A×B are not the same as the ordered pairs in B×A because (orange, 1)≠(1, orange).

In general, if a set A has m elements and a set B has n elements, then the number of ordered pairs in A×B will be m×n. In Example 1, set A contains 3 elements and set B contains 2 elements. Notice that A×B contains 3×2 or 6 ordered pairs.

The set ℝ×ℝ={(x,y):x,y∈ℝ} should be very familiar. It can be viewed as the set of points on the Cartesian plane, and is drawn in Figure 2(a).
The set ℝ×ℕ={(x,y):x∈ℝ,y∈ℕ} can be regarded as all of the points on the Cartesian plane whose second coordinate is a natural number. This is illustrated in Figure 2(b), which shows that ℝ×ℕ looks like infinitely many horizontal lines at integer heights above the x axis. The set ℕ×ℕ can be visualized as the set of all points on the Cartesian plane whose coordinates are both natural numbers. It looks like a grid of dots in the first quadrant, as illustrated in Figure 2(c).

 

Figure 2. Drawings of some Cartesian products

It is even possible for one factor of a Cartesian product to be a Cartesian product itself, as in ℝ×(ℕ×ℤ)={(x,(y,z)):x∈ ℝ,(y,z)∈ℕ×ℤ}.

⛲ Ex2. Sketch the following sets of points in the xy plane.
①. {(x,y)∶x∈[-1,1],y=1}

②. {(x,y)∶|x|=2,y∈[0,1]}

③. {(x,y)∶x∈[1,2],y∈[1,2]}

⛲ Ex3. Sketch these Cartesian products on the xy plane ℝ².
①. {1,2,3}×{-1,0,1}

②. ℕ×ℤ

③. {1}×[0,1]

④. {1,1.5,2}×[1,2]

⑤. [0,1]×[0,1]

⛲ Ex4. Sketch the sets X=[1,3]×[1,3] and Y=[2,4]×[2,4] on the plane ℝ². On separate drawings, shade in the sets X∪Y, X∩Y, X–Y and Y–X. (Hint: X and Y are Cartesian products of intervals.
Read this link with new tab opens, if you want to remember about
🌈 How to Write Subsets of ℝ as Intervals?

We can also define Cartesian products of three or more sets by moving beyond ordered pairs. An ordered triple is a list (x,y,z). The Cartesian product of the three sets ℝ, ℕ and ℤ is ℝ×ℕ×ℤ={(x,y,z):x∈ ℝ,y×ℕ,z×ℤ}.

⛲ Ex5. Sketch this Cartesian products [0,1]×[0,1]×[0,1] on the xy plane ℝ³.

Of course there is no reason to stop with ordered triples. In general,

A₁×A₂×…×A_n={(x₁,x₂,…,x_n):x_i∈A_i for each i=1,2,…,n}.

Be mindful of parentheses. There is a slight difference between ℝ×(ℕ×ℤ) and ℝ×ℕ×ℤ. The first is a Cartesian product of two sets; its elements are ordered pairs (x,(y,z)). The second is a Cartesian product of three sets; its elements look like (x,y,z). To be sure, in many situations there is no harm in blurring the distinction between expressions like (x,(y,z)) and (x,y,z), but for now we consider them as different.

We can also take Cartesian powers of sets. For any set A and positive integer n, the power Aⁿ is the Cartesian product of A with itself n times:

Aⁿ=A×A×…×A={(x₁,x₂,…,x_n):x₁,x₂,…,x_n∈A}In this way, ℝ² is the familiar Cartesian plane and ℝ³ is three-dimensional space. You can visualize how, if ℝ² is the plane, then ℤ²={(m,n):m,n∈ℤ} is a grid of points on the plane. Likewise, as ℝ³ is 3-dimensional space, ℤ³={(m,n,p):m,n,p∈ℤ} is a grid of points in space.
🌈 How to Write Intervals in set-builder form?