If A and B are two non-empty sets then the set of all possible
ordered pairs (x, y) such that the first component x is an element of A and the
second component y is an element of B is called the **Cartesian Product** of set A
and B. It is denoted by **A × B** which is read as “A cross B”.

In the set-builder form, we can write

A × B = {(a, b): a ∈ A, b ∈ B}

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**For example**:

1.
If A = {a, b} and B = {1,
2} then, A × B = {(a, 1), (a, 2), (b, 1), (b, 2)} and B × A = {(1, a), (1, b),
(2, a), (2, b)}.

2.
If A = {1, 2, 3} and B =
{2, 4, 6} then, A × B = {(1, 2), (1, 4), (1, 6), (2, 2), (2, 4), (2, 6), (3,
2), (3, 4), (3, 6)} and B × A = {(2, 1), (2, 2), (2, 3), (4, 1), (4, 2), (4,
3), (6, 1), (6, 2), (6, 3)}.

3.
If A = {H, T} then A × A =
{(H, H), (H, T), (T, H), (T, T)}.

**Note**:

1.
A × B ≠ B × A.

2.
If m = number of elements
of A and n = number of elements of B, then the number of elements of A × B is
mn.

**Representation of the
Cartesian Product**

If A = {1, 2, 3} and B = {1, 2} then the Cartesian Product A × B
can be represented as follows:

(a) __Set of Ordered Pairs__: A × B = {(1, 1), (1, 2), (2, 1,
(2, 2), (3, 1), (3, 2)}

(b) __Lattice Diagram__:

(c) __Table__:

(d) __Mapping Diagram__:

(e) __Tree Diagram__:

**Worked Out Examples**

**Example 1:** If A = {a, b} and B = {1, 2, 3} then find A × B.

**Solution:** Here, A = {a, b} and B = {1, 2, 3}

∴ A × B = {(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)}

**Example 2:** If A = {x ∈ N : x ≤ 4}, find A × A.

**Solution:** Here, A = {x ∈ N : x ≤ 4} ∴ A = {1, 2, 3, 4}

Now, from the above table

A × B = {(1, 1), (a, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3),
(2, 4), (3, 1), (3, 2), (3, 3), (3, 4), (4, 1), (4, 2), (4, 3), (4, 4)}

**Example 3:** Draw the arrow diagram of A × B and B × A if A = {a, b} and B = {x,
y, z}.

**Solution:** Here, A = {a, b} and B = {x, y, z}

A × B in mapping diagram

A × B = {(a, x), (a, y), (a, z), (b, x), (b, y), (b, z)}

B × A in mapping diagram

B × A = {(x, a), (x, b), (y, a), (y, b), (z, a), (z, b)}

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