Representation of Cartesian Product of two Sets

Cartesian product is primarily used in set theory and to represent varied real-life objects, such as a deck of cards and chess boards. The ordered product of two sets is called a Cartesian product. It has an ordered product, which means the resulting pair is arranged in a way that the first element of pairs is derived from the first set and the second element of pairs is derived from the second set. The ordered pair is used for deriving a new set from the already existing sets A and B. Read with us to know more about the properties of the Cartesian product and the representation of the Cartesian product of two sets. 

Understanding Cartesian Product

A Cartesian product is an ordered product of two sets. The word Cartesian came from René Descartes, the famous French mathematician. The steps to find out the Cartesian product are: 

1. In two sets, C (x, y, z) and D (1, 2, 3), take the first element of the set C and the first element of the set D. 
2. Multiply these elements to get the first ordered pair (x, 1). 
3. Take the first element from the set C and the second element from the set D, to form the second ordered pair (x, 2). 
4. Take the first element from set C and the third element from set D to form the third ordered pair (x, 3). 
5. Take the second element from set C and multiply it with all three elements of set D. 
6. Take the third element from set C and multiply it with all three elements of set D. 
7. The entire collection of the ordered pairs from C x D will be {(x, 1), (x, 2), (x, 3), (y, 1), (y, 2), (y, 3), (z, 1), (z, 2), (z, 3)}.
8. We can similarly find all the ordered pairs of D x C following these steps. 

Let us take an example of two sets and find their Cartesian product to understand it better. Let there be a set A with three colours and set B with two objects: 

A = {white, black, red}

B = {p, b} where p stands for a pen and b stands for a ball. 

We can make a total of 6 pairs by multiplying these sets. 

(white, p), (white, b), (black, p), (black, b), (red, p), and (red, p) are the ordered pairs that present the Cartesian products of the set A and B. 

Thus, the Cartesian products of 2 sets A and B are given by A x B. We can define it as a collection of all ordered pairs (a, b) so that a ∈ A and b ∈ B. 

Thus, A X B = {(a, b) : a ∈ A, b ∈ B}

The Cartesian product is also known as the set direct product, cross product, and the product set of A and B. 

Besides the Cartesian product of two sets, there can also be a Cartesian product of more than two sets. Let us look at an example of a Cartesian product of 3 sets. 

Let the 3 sets be A (1, 2), B(3, 4) and C(5, 6). 

The total number of ordered pairs will be A X B X C = 2 X 2 X 2 = 8

A × B × C = {1, 2} × {3, 4} × {5, 6} 

This will give us {(1, 3, 5), (1, 3, 6), (1, 4, 5), (1, 4, 6), (2, 3, 5), (2, 3, 6), (2, 4, 5), (2, 4, 6)}. 

Further, let us now derive the Cartesian product of an empty set. An empty set is an exceptional set that has no elements. The Cartesian product of an empty set with another will give a resultant empty set. For example, let us take set C = {1, 2} and set D = ϕ.

Here C × D = ϕ 

Similarly, D × C = ϕ

Cartesian Product Properties

There are some Cartesian set properties that you must remember, such as: 

1. The Cartesian product is non-commutative, meaning that the result of the product is dependent on the order of the sets. 

That means C X D ≠ D X C

C X D = D X C only when C = D

2. The Cartesian product is non-associative, meaning that rearranging the position will change the result of the Cartesian product. 

(C X D) X E ≠ C X (D X E) 

3. In the Cartesian product, there is a distributive property over the intersection of the set. 

A × (B ∩ C) = (A × B) ∩ (A × C)

4. In the Cartesian product, there is a distributive property over the union of the set. 

A × (B ∪ C) = (A × B) ∪ (A × C)

Cartesian Product Cardinality 

The cardinality of a given set is described as the total number of elements in the set. The cardinal number of set A is given by n(A) that equals the number of total elements present in set A. For instance, the cardinal number of a set A having all English alphabets will be n(A) = 26. 

Similarly, we can also find the cardinality of the Cartesian product. The cardinality of the Cartesian product of 2 sets equals the products of the cardinalities of both sets. 

n(C × D) = n(D × C) = n(C) × n(D)

For example, let us take two sets C {2,3} and D {5,4,7}. 

Here, n(C) = 2 and n(D) = 3

We know that n(C × D) = n(C) × n(D)

⇒ n(C × D) = 2 × 3 = 6