CARTESIAN PRODUCT OF TWO SETS

About "Cartesian product of two sets"

Cartesian product of two sets :

If A and B are two non-empty sets, then the set of all ordered pairs (a, b) such that "a" belongs to A and "b" belongs to B, is called the Cartesian product of A and B, to be denoted by A x B.

Thus,

A x B  =  { (a, b) : a ∈ A and b ∈ B }

And the Cartesian product of B and A, to be denoted by       B x A.

Thus, 

B x A  =  { (b, a) : a ∈ A and b ∈ B }

And also,

A x B  ≠  B x A 

But, 

n(A x B)  =  n(B x A) 


If A is null set or B is null set, we define that A x B is null set.  

The figure given below clearly illustrates the Cartesian product of two sets.

Cartesian product of two sets - Examples

Example 1 :

Let A  = { 1, 2, 3 }, B  =  { 4, 5 }. Find A x B and B x A

Solution : 

A x B  =  { (1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5) }

B x A  =  { (4, 1), (4, 2), (4, 3), (5, 1), (5, 2), (5, 3) }

Example 2 :

If A x B  =  { (3, 2), (3, 4), (5, 2), (5, 4) }, find the sets A and B.

Solution : 

Clear, A is the set of all first co-ordinates of A x B, while B is the set of all second co-ordinates of elements of A x B. 

Therefore A  =  { 3, 5 } and B  =  { 2, 4 }

Example 3 :

Let P  = { 1, 3, 6 }, Q  =  { 3, 5 }

Then, the product set 

P x Q  =  { (1, 3), (1, 5), (3, 3), (3, 5), (6, 3), (6, 5) }

Q x P  =  { (3, 1), (3, 3), (3, 6), (5, 1), (5, 3), (5, 6) }

Notice that n(P) = 3, n(Q) = 2, n(PxQ) = 6 and n(QxP) = 6

And also,

n(PxQ)  =  n(P) x n(Q)  

n(QxP)  =  n(P) x n(Q)  

In Cartesian product, the ordered pairs (3, 5) and (5, 3) are not equal.

So,

P x Q  ≠  Q x P 

But,

n(P x Q)  =  n(Q x P)

After having gone through the stuff given above, we hope that the students would have understood "Cartesian product of sets". 

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