# Cartesian product

## Cartesian product

$$A\times B=\left\{\left(a,b\right):a\in A,b\in B\right\}$$

The Cartesian Product of two sets **A** and **B** is the set of all Ordered Pairs **(a,b)** where the first element of order pairs **a** belongs to first set **A** and second element of ordered pairs **b** belongs to second set **B**.

Example:

$A=\{1,3,6\},B=\{x,y\}$

$AxB=\left\{\left(1,x\right),\left(1,y\right),\left(3,x\right),\left(3,y\right),\left(6,x\right),\left(6,y\right)\right\}$

## Cartesian square

$${A}^{2}=AxA=\left\{\left(a,b\right):a\in A,b\in B\right\}$$

The Cartesian squere of set **A** is the set of all Ordered Pairs **(a,b)** where both elements of order pairs belongs to first set **A**.

Example:

$A=\{1,3,6\}$

$AxA=\left\{\left(1,1\right),\left(1,3\right),\left(1,6\right),\left(3,1\right),\left(3,3\right),\left(3,6\right),\left(6,1\right),\left(6,3\right),\left(6,6\right)\right\}$

## Properties of Cartesian Product

$$\left|A\right|=m,\left|B\right|=n\Rightarrow \left|AxB\right|=m\xb7n$$

$$A=\varnothing \vee B=\varnothing \Rightarrow AxB=\varnothing $$

Keywords: sets, cartesian product, cartesian square