Cartesian Product of Two Sets | Cross Product of Sets

Let us consider A and B to be two non-empty sets and the Cartesian Product is given by AxB set of all ordered pairs (a, b) where a ∈ A and b ∈ B.

AxB = {(a,b) | a ∈ A and b ∈ B}. Cartesian Product is also known as Cross Product.

Consider Set A = { 3, 4, 5} B = {x, y} then AxB is given by

A

3 4 5

B

x

y

AxB = {(3,x), (4,x), (5, x), (3, y), (4, y), (5, y)}

In the same way, we can find the value of BxA

BxA = {(x,3), (x, 4), (x, 5), (y, 3), (y, 4), (y, 5)}

Thus from the example, we can say that AxB and BxA don’t have the same ordered pairs. Therefore, AxB ≠ BxA.

If A = B then AxB is called the Cartesian Square of Set A and is represented as A².

A² = {(a,b) a ∈ A and b ∈ A}

Solved Examples

1. If A = {3, 4, 5} B = {1, 2} find the value of AxB, BxA, A², B²?

Solution:

Given A = {3, 4, 5} B = {1, 2}

AxB = {3, 4, 5}x{1, 2}

= {(3,1), (3, 2), (4, 1), (4, 2), (5, 1), (5, 2)}

BxA = {1, 2}x{3, 4, 5}

= {(1, 3), (1,4), (1,5), (2, 3), (2, 4), (2,5)}

A² = {3, 4, 5}x{3, 4, 5}

= {(3, 3), (3, 4), (3, 5), (4, 3), (4, 4), (4, 5), (5, 3), (5, 4), (5, 5)}

B² = {1, 2}x{ 1, 2}

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

2. If A = {x, y,z} then B = {y, z} find the Cartesian Product AxB?

Solution:

A = {x, y,z}

B = {y, z}

AxB = {x, y,z}x{y, z}

= {(x,y), (x,z), (y, y), (y, z), (z, y), (z, z)}

3. If A = { 4, 5, 6} B = {7, 8} find the Cartesian Product of AxB?

Solution:

A = { 4, 5, 6}

B = {7, 8}

AxB = {4, 5, 6}x{7, 8}

AxB = {(4,7), (4, 8), (5,7), (5, 8), (6, 7), (6, 8)}