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".

Apart from the stuff given above, if you want to know more about "Cartesian product of sets", please click here

If you need any other stuff in math, please use our google custom search here.