Cartesian Products of Sets

In mathematics, you may come across several relations such as number p is greater than number q, line m parallel to line n, set A subset of set B, etc. In all these, we can notice a relationship that involves pairs of objects in a specific order. Also, you might have learned different set operations in maths. Here, you will learn how to link pairs of elements from two sets and then introduce relations between the two elements in pairs.

Cartesian Products of Sets Definition

In this section, you will learn the definition for the Cartesian products of sets with the help of an illustrative example. Let A and B be the two sets such that A is a set of three colours of tables and B is a set of three colours of chairs objects, i.e.,

A = {brown, green, yellow}

B = {red, blue, purple},

Let’s find the number of pairs of coloured objects that we can make from a set of tables and chairs in different combinations. They can be paired as given below:

(brown, red), (brown, blue), (brown, purple), (green, red), (green, blue), (green, purple), (yellow, red), (yellow, blue), (yellow, purple)

There are nine such pairs in the Cartesian product since three elements are there in each of the defined sets A and B. The above-ordered pairs represent the definition for the Cartesian product of sets given. This product is denoted by “A × B”.

Cartesian Product of Sets Formula

Given two non-empty sets P and Q. The Cartesian product P × Q is the set of all ordered pairs of elements from P and Q, i.e.,

P × Q = {(p,q) : p ∈ P, q ∈ Q}

If either P or Q is the null set, then P × Q will also be an empty set, i.e., P × Q = φ

What is the Cardinality of Cartesian Product?

The cardinality of Cartesian products of sets A and B will be the total number of ordered pairs in the A × B.

Let p be the number of elements of A and q be the number of elements in B.

So, the number of elements in the Cartesian product of A and B is pq.

i.e. if n(A) = p, n(B) = q, then n(A × B) = pq.

How to Find the Cartesian Products?

As defined above, the Cartesian product A × B between two sets A and B is the set of all possible ordered pairs with the first element from A and the second element from B. In this section, you will learn how to find the Cartesian products for two and three sets, along with examples.

Let’s have a look at the example given below. Here, set A contains three triangles of different colours and set B contains five colours of stars.

The Cartesian product of given sets A and B is given as a combination of distinct colours of triangles and stars. Thus, a total of 15 pairs are formed in A × B from the given sets.

Cartesian Products of Two Sets

For example, A = {a₁, a₂, a₃} and B = {b₁, b₂, b₃, b₄} are two sets. The Cartesian product of A and B can be shown as:

Cartesian Products of Three Sets

Suppose A be a non-empty set and the Cartesian product A × A × A represents the set A × A × A ={(x, y, z): x, y, z ∈ A} which means the coordinates of all the points in three-dimensional space. This forms the basis for the Cartesian product of three sets. The below example helps in understanding how to find the Cartesian product of 3 sets.

Example:

Find the Cartesian product of three sets A = {a, b}, B = {1, 2} and C = {x, y}.

Solution:

Given,

A = {a, b}

B = {1, 2}

C = {x, y}

The ordered pairs of A × B × C can be formed as given below:

1st pair ⇒ {a, b} × {1, 2} × {x, y} ⇒ (a, 1, x)

2nd pair ⇒ {a, b} × {1, 2} × {x, y} ⇒ (a, 1, y)

3rd pair ⇒ {a, b} × {1, 2} × {x, y} ⇒ (a, 2, x)

4th pair ⇒ {a, b} × {1, 2} × {x, y} ⇒ (a, 2, y)

5th pair ⇒ {a, b} × {1, 2} × {x, y} ⇒ (b, 1, x)

6th pair ⇒ {a, b} × {1, 2} × {x, y} ⇒ (b, 1, y)

7th pair ⇒ {a, b} × {1, 2} × {x, y} ⇒ (b, 2, x)

8th pair ⇒ {a, b} × {1, 2} × {x, y} ⇒ (b, 2, y)

Thus, the ordered pairs of A × B × C can be written as:

A × B × C = {(a, 1, x), (a, 1, y), (a, 2, x), (a, 2, y), (b, 1, x), (b, 1, y), (b, 2, x), (b, 2, y)}

Cartesian Products of Sets Properties

Some of the important properties of Cartesian products of sets are given below.

(i) Two ordered pairs are equal, if and only if the corresponding first elements are equal and the second elements are also equal.

(ii) If there are m elements in A and n elements in B, then there will be mn elements in A × B. That means if n(A) = m and n(B) = n, then n(A × B) = mn.

(iii) If A and B are non-empty sets and either A or B is an infinite set, then A × B is also an infinite set.

(iv) A × A × A = {(a, b, c) : a, b, c ∈ A}. Here (a, b, c) is called an ordered triplet.

(v) The Cartesian product of sets is not commutative, i.e. A × B ≠ B × A

(vi) The Cartesian product of sets is not associative, i.e. A × (B × C) ≠ (A × B) × C

(vii) If A is a set, then A × ∅ = ∅ and ∅ × A = ∅.

(viii) If A and B are two sets, A × B = B × A if and only if A = B, or A = ∅, or B = ∅.

(ix) Let A, B and C be three non-empty sets, then,

• A × (B ∩ C) = (A × B) ∩ (A × C)
• A × (B ∪ C) = (A × B) ∪ (A × C)
• (A ∩ B) × C = (A × C) ∩ (B × C)
• (A ∪ B) × C = (A × C) ∪ (B × C)

Solved Examples

Example 1: 

If  A = {1, 2, 3} and B = {3, 4}, find the Cartesian product of A and B.

Solution:

Given,

A = {1, 2, 3}

B = {3, 4}

The Cartesian product of A and B =  A × B

= {1, 2, 3} × {3, 4}

= {(1, 3), (1, 4), (2, 3), (2, 4), (3, 3), (3, 4)}

Example 2:

If P = {5, 6}, form the set P × P × P.

Solution:

P × P × P = {5, 6} × {5, 6} × {5, 6}

= {(5, 5, 5), (5, 5, 6), (5, 6, 5), (5, 6, 6), (6, 5, 5), (6, 5, 6), (6, 6, 5), (6, 6, 6)}

Example 3: 

The Cartesian product A × A has 9 elements, among which are found (–1,  0) and (0, 1). Find the set A and the remaining elements of A × A.

Solution:

As we know, if n(A) = p and n(B) = q, then n(A x B) = pq

Thus, n(A x A) = n(A) x n(A)

Given,

n(A x A) = 9

n(A) x n(A) = 9

⇒ n(A) = 3….(i)

Also, given that (- 1, 0) and (0, 1) are two of the nine ordered pairs of A x A.

We know that A x A = {(a, a) : a ∈ A} .

Therefore, – 1, 0, and 1 are the elements of A…..(ii)

From (i) and (ii),

A = {-1, 0, 1}

Hence, the remaining elements of set A x A are (- 1, – 1), (- 1, 1), (0, – 1), (0, 0), (1, – 1), (1, 0), and (1, 1).

Video Lesson on What are Sets

Cartesian Products of Sets Questions

Go through the below sets questions based on the Cartesian product.

1. If A = {3, 4, 5}, B = {5, 6} and C = {6, 7, 8}, then find the following.

(i) A × (B ∩ C) (ii) (A × B) ∩ (A × C) (iii) A × (B ∪ C) (iv) (A × B) ∪ (A × C)

1. If X = {2, 3}, then form the set X × X × X.
2. If A × B = {(a, x),(a , y), (b, x), (b, y)}, then find set A and set B.
3. Let A and B be two sets such that n(A) = 3 and n(B) = 2. If (x, 1), (y, 2), (z, 1) are in A × B, find A and B, where x, y and z are distinct elements.

Download BYJU’S – The Learning App and get engaging videos to learn maths concepts effectively.