Cartesian Product

Cartesian Product

## Definition / Concept

The Cartesian product of $A$ and $B$ is the set of all ordered pairs whose first component belongs to $A$ and second component belongs to $B$.

## Formula

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

If:

$$n(A)=m,\qquad n(B)=n$$

then:

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

## Terminologies

- Ordered pair → Pair $(a,b)$ in which the order of components matters.

- Cartesian product → Set of all ordered pairs formed from two sets.

## Conditions / Special Cases

In general:

$$A\times B\ne B\times A$$

If either set is empty:

$$A\times\varnothing=\varnothing$$

$$\varnothing\times A=\varnothing$$

For three sets:

$$A\times B\times C=\{(a,b,c):a\in A,\ b\in B,\ c\in C\}$$

If:

$$n(A)=m,\quad n(B)=n,\quad n(C)=p$$

then:

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

## Usage

- Forms the basis for relations and functions.

- Used in ordered-pair and counting problems.