Cartesian Product of Two Sets - Definition and Step-by-Step Examples

The Cartesian Product of two sets is the set of all possible ordered pairs where the first element comes from the first set and the second element comes from the second set.

Definition

If $A$ and $B$ are two sets, then the Cartesian Product of $A$ and $B$ , denoted by $A \times B$ , is defined as:

$A \times B = {(a, b) ∣ a \in A, b \in B}$

Each element of $A \times B$ is an ordered pair $(a, b)$ , where: - $a$ belongs to set $A$ - $b$ belongs to set $B$

Important Notes

The order of sets matters: $A \times B$ is generally not equal to $B \times A$ .
If one of the sets is empty, the Cartesian Product is also empty.

Step-by-Step Example 1

Let $A = {1, 2}$ $B = {x, y}$

Step 1: Identify all elements of $A$ and $B$ . - $A$ has elements 1 and 2 - $B$ has elements $x$ and $y$

Step 2: Form ordered pairs by taking each element from $A$ and pairing it with each element from $B$ :

$A \times B = {(1, x), (1, y), (2, x), (2, y)}$

Step-by-Step Example 2

Let $C = {a, b}$ $D = {1, 2, 3}$

Step 1: Elements of $C$ : $a$ , $b$ Step 2: Elements of $D$ : 1, 2, 3

Step 3: Make all ordered pairs:

$C \times D = {(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)}$

Example 3: Cartesian Product with Itself

Let $E = {0, 1}$

Then $E \times E = {(0, 0), (0, 1), (1, 0), (1, 1)}$

This is useful in representing points in a 2D grid.

Visual Meaning

In coordinate geometry, $A \times B$ gives all possible coordinate points where the x-coordinate comes from set $A$ and the y-coordinate from set $B$ .

Summary

The Cartesian Product combines elements from two sets into ordered pairs.
It's used in coordinate geometry, databases, and relation mappings.
Always pay attention to the order of sets.