### Theory:

Recalling finite and infinite set:
A set with a finite number of elements is called a finite set.
Example:
1. The set of members in a classroom.
2. The set of letters in Tamil alphabets.
3. The collection of movies seen.
4. The set of amusement park you visited.
These are countable. So the sets are finite.
A set with an infinite number of elements is called an infinite set.
Example:
1. The set of all multiples of $$5$$.
2. The set of all points on a line.
3. The set $A\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\left\{0,\phantom{\rule{0.147em}{0ex}}1,2,\phantom{\rule{0.147em}{0ex}}3,...\right\}$.
4. The set of rational numbers between $$-3$$ and $$-2$$.
5. The set of all prime numbers.
These sets have an infinite number of elements. Thus, these are called an infinite set.
Important!
Some infinite sets cannot be expressed in the roster form.
Example:
The set of rational  numbers between $$10$$ and $$11$$ cannot be expressed in roaster form, because there is an infinite number of rational numbers between two rational numbers.
When a set is finite, we always eagerly like to know how many elements it has. The cardinal number gives the number of elements in a set.
The number of elements in a set is called the cardinal number of a set. The cardinality of set $$A$$ is denoted by $$n(A)$$.
Example:
Consider the set of vowels in English alphabets.

$A\phantom{\rule{0.147em}{0ex}}=\left\{a,\phantom{\rule{0.147em}{0ex}}e,\phantom{\rule{0.147em}{0ex}}i,\phantom{\rule{0.147em}{0ex}}o,\phantom{\rule{0.147em}{0ex}}u\right\}$

The number of elements in set $$A$$ is $$5$$.

Thus, the cardinality of set $$A$$ that is $$n(A) = 5$$.
Example:
Now consider the set $B=\left\{1,\left\{2,3\right\},\left\{4\right\},\left\{2,4\right\},\left\{1,2,3\right\}\right\}$

We have to find the number of distinct elements in set $$B$$.

The distinct elements in set $$B$$ are $$1, 2, 3$$, and $$4$$.

The number of elements in set $$B$$ is $$4$$.

Thus, the cardinality of set $$B$$ is $$n(B) = 4$$.
Important!
Note that we should not count the repeated elements.