Cartesian Products of Sets

Here we will learn how to link pairs of elements from two sets and then introduce relations between the two elements in pairs.

Cartesian Products of Sets Definition

Let A and B be the two sets such that A is a set of three colours of tables and B is a set of three colours of chairs objects, i.e.,

$A=\{$brown, green, yellow$\}$

$B=\{$red, blue, purple$\}$

Let’s find the number of pairs of coloured objects that we can make from a set of tables and chairs in different combinations. They can be paired as given below:

(brown, red), (brown, blue), (brown, purple), (green, red), (green, blue), (green, purple), (yellow, red), (yellow, blue), (yellow, purple)

There are nine such pairs in the Cartesian product since three elements are there in each of the defined sets $A$ and $B$. The above-ordered pairs represent the definition for the Cartesian product of sets given. This product is denoted by $AXB$.

Cartesian Product of Sets Formula

Given two non-empty sets $P$ and $Q$. The Cartesian product $PXQ$ is the set of all ordered pairs of elements from $P$ and $Q$ ,i.e.,

$PXQ=\{(p,q):p\in P,q\in Q\}$

If either $P$ or $Q$ is the null set, then $PXQ$ will also be an empty set, i.e., $PXQ=\emptyset$

What is the Cardinality of Cartesian Product?

The cardinality of Cartesian products of sets $A$ and $B$ will be the total number of ordered pairs in the $AXB$

Let $p$ be the number of elements of $A$ and $q$ be the number of elements in $B$.

So, the number of elements in the Cartesian product of $A$ and $B$ is $pq$

i.e. if $n(A)=p,n(B)=q$ then $n(AXB)=pq$