Cartesian Product of Sets

Let us take two sets A & B

A = {Red, Blue}

B = {Bag, Shirt, Jeans}

Now, how many pairs can we have?

We can have

(Red, Bag), (Red, Shirt) , (Red, Jeans)

and

(Blue, Bag), (Blue, Shirt), (Blue, Jeans)

Cartesian product is the set of all these pairs.

So, we write

A × B = {(Red, Bag), (Red, Shirt) , (Red, Jeans),
        (Blue, Bag), (Blue, Shirt), (Blue, Jeans)}

So, definition of Cartesian Product is

For set A & B

A × B = {(a, b): a ∈ A, b ∈ b}

(a, b) is called ordered pair .

Note that:

(a, b) ≠ (b, a)

Let us take some examples

Let A = {1, 2, 3}, B = {3, 4} and C = {4, 5, 6}

Find A × B

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

A  × B = {(1, 3), (1, 4),

        (2, 3), (2, 4),

        (3, 3), (3, 4)}

Find B × A

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

B × A = { (3, 1), (3, 2), (3, 3),

        (4, 1), (4, 2), (4, 3)}

Note that

A × B ≠ B × A

Find A × C

A = {1, 2, 3}, C = {4, 5, 6}

A × C = {(1, 4), (1, 5), (1, 6),

       (2, 4), (2, 5), (2, 6),

       (3, 4), (3, 5), (3, 6)}

Find C × A

C = {4, 5, 6}, A = {1, 2, 3}

C × A = {(4, 1), (4, 2), (4, 3),

(5, 1), (5, 2), (5, 3),

(6, 1), (6, 2), (6, 3), }

Note that

A × C ≠ C × A

Find B × C

B = {3, 4}, C = {4, 5, 6}

B × C = {(3, 4), (3, 5), (3, 6),

        (4, 4), (4, 5), (4, 6)}

Find C × B

C = {4, 5, 6}, B = {3, 4}

C × B = {(4, 3), (4, 4),

(5, 3), (5, 4),

(6, 3), (6, 4)}

Note that

B × C ≠ C × B