Cartesian Product and Ordered Pairs

Before understanding Cartesian product definition, one needs to have a clear knowledge of what exactly is Cartesian? This particular term is used while plotting a graph on the 2D axis. It comprises abscissa or the x-axis and ordinates that is the y-axis. Hence, (1, 2) in the graph describes the position of a point with 1 being the x-axis or abscissa and 2 being the y-axis or ordinate. Adding the term product with Cartesian means the product of these elements in an orderly manner. The idea behind this section is to make you aware of the Cartesian product and Ordered pair definition. 

So, what is a Cartesian Product? 

The best way to put the Cartesian product and ordered pairs definition is: the collection of all the ordered pairs that can be obtained through the product of two non-empty sets. So, if we take two non-empty sets, then an ordered pair can be formed by taking elements from the two sets. Given that, the first element of the pair belongs from set A while the second one from set B. Cartesian product gives us the collection of all the pairs. 

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Cartesian Product Formula

If we take two non-empty sets A and B, then the Cartesian product is denoted by A x B, where all the ordered pairs (a, b) are made in a way that a ∈ A and b ∈ B. So, the formula for the Cartesian product is given by: 

A×B = {(a, b): a ∈ A, b ∈B} 

Sometimes the Cartesian product of two sets is also known as a cross-product or just product of the sets A and B. 

Interesting Facts About Cartesian Product

While looking into the Cartesian product example you will find some of the interesting facts about a Cartesian product that are worth mentioning:  

  • |A x B| = |A|*|B|

Bear in mind that it is only applicable if both A and B are finite sets, for there are |A| number of choices for the first component whereas |B| number of choices for the second component in the ordered pairs. 

  • A×B ≠ B×A

It is only applicable when A ≠ B. This particular proposition can be satisfied with the reason that both A and B are non-empty sets. Suppose there is an element x which is present in A but not in B. Then A x B contains an ordered pair with x as the first component which will not be present in B x A.  

  • For Every Set A, A× ∅ = ∅ and ∅ × A = ∅

Where ∅ defines an empty set.  

  • If A, B and C are sets, then: 

1. A x (B ∩ C) = (A x B) ∩ (A x C)

2. A x (B ∪ C) = (A x B) ∪ (A x C)

3. (A∩ B) x C = ( A x C) ∩ ( B x C)

4. (A ∪ B) x C = (A x C) ∪ (B x C) 

Cartesian Product and Ordered Pairs Examples

Here is a cartesian product of sets example:

Example 1

Let there be two sets, A = {1, 2, 3} and B = {x, y}. Then find the Cartesian product of the two sets in: 

  1. A x B

  2. B x A

  3. A x A

  4. B x B 

Solution

According to the definition of the Cartesian product:  

  1. A x B = {(1, x), (2, x), (3, x), (1, y), (2, y), (3, y)}

  2. B x A = {(x, 1), (y, 1), (x, 2), (y, 2), (x, 3), (y, 3)}

  3. A x A = {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}

  4. B x B = {(x, x), (x, y), (y, x), (y, y)}

 

Example 2 

Let us take X and Y as two non-empty sets. Some of the ordered pairs are given as (a, 1), (b, -2), (x, 1), (y, -2). Find X and Y, if a, b, x, y are distinct elements. Given: n(X) = 4 and n(Y) = 2.  

Solution:

 X = {a, b, x, y} and Y = {1, -2}

FAQ (Frequently Asked Questions)

1. What are the Implementations of Cartesian Products?

The Cartesian product was invented by René Descartes. In fact, the name Cartesian product has also been derived from the same person. The idea of the Cartesian product originated from analytical geometry, which is now conceptualized in the general term as a direct product. As it is derived from the definition of ordered pairs, the Cartesian products are vigorously implemented within the set theory. As for the real-world application, the same A x B can be seen while dealing with databases. In fact, Cartesian products form the building blocks of the modern world of data management software, starting from SQL to Python. A Cartesian product has several applications in the field of mathematics too. 

2. Is the Cartesian Product Only Constrained to Two Sets?

No. The Cartesian product of sets goes beyond the limitation of two sets. For example, if A, B and C are three sets, then every element for the Cartesian product of A x B x C can be represented by (a, b, c). It is just that the complexity of the computation increases as the number of sets goes on increasing. That is why, all the examples within the article are shown with two sets for better understanding. 

For example, if A = (a1, a2), B = (b1, b2) and C = (c1, c2)

Then A x B x C = {(a1, b1, c1), (a1, b1, c2), (a1, b2, c1), (a1, b2, c2), (a2, b1, c1), (a2, b1, c2), (a2, b2, c1), (a2, b2, c2)}

The total number of elements in A x B x C is given by n(A) * n(B) * n(C)