Cartesian Product of Two Sets

If A and B are two non-empty sets, then their Cartesian product A × B is the set of all ordered pair of elements from A and B.

A × B = {(x, y) : x ∈ A, y ∈ B}

Suppose, if A and B are two non-empty sets, then the Cartesian product of two sets, A and set B is the set of all ordered pairs (a, b) such that a ∈A and b∈B which is denoted as A × B.

For Example;

1. If A = {7, 8} and B = {2, 4, 6}, find A × B.

Solution:

A × B = {(7, 2); (7, 4); (7, 6); (8, 2); (8, 4); (8, 6)}

The 6 ordered pairs thus formed can represent the position of points in a plane, if a and B are subsets of a set of real numbers.

2. If A × B = {(p, x); (p, y); (q, x); (q, y)}, find A and B.

Solution:

A is a set of all first entries in ordered pairs in A × B.

B is a set of all second entries in ordered pairs in A × B.

Thus A = {p, q} and B = {x, y}

3. If A and B are two sets, and A × B consists of 6 elements: If three elements of A × B are (2, 5) (3, 7) (4, 7) find A × B.

Solution:

Since, (2, 5) (3, 7) and (4, 7) are elements of A × B.

So, we can say that 2, 3, 4 are the elements of A and 5, 7 are the elements of B.

So, A = {2, 3, 4} and B = {5, 7}

Now, A × B = {(2, 5); (2, 7); (3, 5); (3, 7); (4, 5); (4, 7)}

Thus, A × B contain six ordered pairs.

4. If A = { 1, 3, 5} and B = {2, 3}, then

Find: (i) A × B (ii) B × A (iii) A × A (iv) (B × B)

Solution:

A ×B={1, 3, 5} × {2,3} = [{1, 2},{1, 3},{3, 2},{3, 3},{5, 2},{5, 3}]

B × A = {2, 3} × {1, 3, 5} = [{2, 1},{2, 3},{2, 5},{3, 1},{3, 3},{3, 5}]

A × A = {1, 3, 5} × {1, 3, 5}= [{1, 1},{1, 3},{1, 5},{3, 1},{3, 3},{3, 5},{5, 1},{5, 3},{5, 5}]

B × B = {2, 3} × {2, 3} = [{2, 2},{2, 3},{3, 2},{3, 3}]

Note:

If either A or B are null sets, then A ×B will also be an empty set, i.e., if A = ∅ or

B = ∅, then A × B = ∅