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.