Cartesian Product

Cartesian Product - Definition, Explanation, and Examples

The Cartesian Product is an operation used in mathematics to combine two sets and form a new set made of ordered pairs. This concept is widely used in set theory, coordinate geometry, and computer science.

Definition

If $A$ and $B$ are two sets, the Cartesian product of $A$ and $B$ is the set of all ordered pairs where:

- The first element is from set $A$ - The second element is from set $B$

It is denoted as:

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

Important Points

• The order in each pair matters. That is, $(a,b)\ne (b,a)$ unless $a=b$.
• If either set is empty, the Cartesian product is also empty:

 $A\times \mathrm{\varnothing }=\mathrm{\varnothing }$ and $\mathrm{\varnothing }\times B=\mathrm{\varnothing }$

• The total number of ordered pairs in $A\times B$ is:

 $n(A\times B)=n(A)\times n(B)$

Step-by-Step Example 1

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

Step 1: Take each element of set $A$ Step 2: Pair it with each element of set $B$

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

There are 4 ordered pairs because $2\times 2=4$.

Step-by-Step Example 2

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

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

$n(C\times D)=2\times 3=6$ ordered pairs.

Example 3: Cartesian Product with Itself

Let: $E=\{0,1\}$

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

This is also known as a set of points in a 2D space — for example, like grid points in coordinate geometry.

Visual Representation

If you consider $A=\{1,2\}$ and $B=\{3,4\}$, then $A\times B$ can be visualized as points in a table or plane:

|| **B = 3** || **B = 4** || |------------|-------------| | (1, 3) | (1, 4) | | (2, 3) | (2, 4) |

Applications

• Coordinate geometry (e.g., the Cartesian plane)
• Relations and functions
• Computer science and databases
• Logic and discrete mathematics

Summary

The Cartesian product of two sets combines all elements from both sets into ordered pairs. It forms the basis of many mathematical concepts like relations, functions, and coordinates.