Cartesian Product of Sets: Class 11 Maths Notes, Definition, Properties & Solved Examples

A Cartesian product is a way to combine elements of two sets to form ordered pairs. An ordered pair means the order of the elements matters. The result is a new set made up of all possible combinations of one item from each set.

Example:
Set $A=\left\{\mathrm{1,2}\right\}$
Set $B=\{x,y\}$

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

Importance of Cartesian Products for Engineering Entrance Exams

• For engineering exams, it is expected that you know the core concepts and theoretical properties of the Cartesian products of sets. 
• You may have to identify the different domains and the ranges of functions, for instance. 
• Usually, in exams like the JEE Mains, the Cartesian products of sets is a foundation for set theory questions.  

In Cartesian products, order is very important. The pair ( $1,\mathrm{x}$ ) is not the same as ( $\mathrm{x},1$ ), because the first element must come from the first set, and the second from the second set.

So:
$-\left(\mathrm{1,2}\right)\ne \left(\mathrm{2,1}\right)-(\mathrm{a},\mathrm{b})\ne (\mathrm{b},\mathrm{a})$ , unless $\mathrm{a}=\mathrm{b}$

$C\times D=\left\{\right(a,b)\mid a\in C,b\in D\}$

This means: for every element $a$  in set $C$ and every element $b$ in set $D$ , we make a pair $(a,b)$ .

Example:
$C=\{\mathrm{x},\mathrm{y},\mathrm{z}\}$

$D=\{\mathrm{1,2},3\}$

$C\times D=\left\{\right(x,1),(x,2),(x,3),(y,1),(y,2),(y,3),(z,1),(z,2),(z,3\left)\right\}$

The cartesian product, also known as the cross-product or the product set of C and D is obtained by following the below-mentioned steps:

The first element $x$ is taken from the set $C\{x,y,z\}$ and the second element 1 is taken from the second set $D\{\mathrm{1,2},3\}$

Both these elements are multiplied to form the first ordered pair ( $\mathrm{x},1$ )
The same step is repeated for all the other pairs too until all the possible combinations are chosen

The entire collection of all such ordered pairs gives us a cartesian product $C\times D=\left\{\right(x,1)$ , $(x,2),(x,3),(y,1),(y,2),(y,3),(z,1),(z,2),(z,3)\}$ .

Similarly, we can find the cartesian product of $\mathrm{D}\times \mathrm{C}$ .

What About $\mathrm{D}\times \mathrm{C}$ ?
$D=\{\mathrm{1,2},3\},C=\{x,y,z\}$

$D\times C=\left\{\right(1,x),(1,y),(1,z),(2,x),(2,y),(2,z),(3,x),(3,y),(3,z\left)\right\}$

Clearly, $\mathrm{D}\times \mathrm{C}\ne \mathrm{C}\times \mathrm{D}$

Cartesian Product of More Than Two Sets

Example:
$A=\left\{\mathrm{2,3}\right\},B=\{x,y\},C=\left\{\mathrm{5,6}\right\}$

$A\times B\times C=\left\{\right(2,x,5),(2,x,6),(2,y,5),(2,y,6),(3,x,5),(3,x,6),(3,y,5),(3,y,6\left)\right\}$

Cartesian Product with Empty Sets

If even one set is empty, the Cartesian product is empty.

Example:

$\mathrm{C}=\left\{\mathrm{1,2}\right\},\mathrm{D}=\mathrm{\varnothing }\mathrm{C}\times \mathrm{D}=\varnothing$
$C\times D=\mathrm{\varnothing }$ and $\mathrm{D}\times \mathrm{C}=\varnothing$

If A and B are countable (even infinite), then A × B is also countable.