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

Thus, A x B = { (a,b) |a ∈ A,b ∈ B }

A x B is the set of all possible ordered pairs between the elements of A and B such that the first coordinate is an element of A and the second coordinate is an element of B.

B x A is the set of all possible ordered pairs between the elements of A and B such that the first coordinate is an element of B and the second coordinate is an element of A.

If a = b, then (a, b) = (b, a).

The 'Cartesian Product' is also referred as 'Cross Product'.

In general

AxB ≠ BxA,

But,

n(A x B) = n(B x A)

AxB = ∅, if and only if A = ∅ or B = ∅.

If n(A) = p and n(B) = q ,then

n(AxB) = pq

**Problem 1 :**

Find AxB , AxA and BxA :

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

**Solution :**

**A = {2, -2, 3} and B = {1, -4}.**

**AxB = {(2, 1), ****(2, -4), ****(-2, 1), ****(-2, -4), ****(3, 1), ****(3, -4)}**

A = {2, -2, 3} and A = {2, -2, 3}.

**AxA = ****{(2, 2), (2, -2), (2, 3), ****(-2, 2), (-2, -2), (-2, 3), **

**(3, 2), (3, -2), (3, 3)}**

B = {1, -4} and A = {2, -2, 3}.

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

**Problem 2 :**

Find AxB , AxA and BxA :

**A = B = {p, q}**

**Solution : **

**A = ****{p, q} and B**** = ****{p, q}**

**AxB = {(p, p), ****(p, q), ****(q, p), ****(q, q)}**

**A = ****{p, q}****, ****A = ****{p, q}**

**AxA =**** {(p, p) (p, q) (q, p) (q, q)****}**

**B = ****{p, q} and B**** = ****{p, q}**

**BxA = ****{(p, p), ****(q, p), ****(p, q), ****(q, q)}**

**Problem 3 :**

Find AxB , AxA and BxA :

**A = {m, n} and ****B = ∅**

**Solution :**

**Because ****B = ∅, **

**AxB = ****∅**

**BxA **= ∅

**A = {m, n} and ****A = {m, n}**

AxA = {(m, m), (m, n), (n, m), (n, n)}

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