Finite and Infinite Sets

Finite sets and Infinite sets are totally different from each other. As the name suggests, the finite set is countable and contains a finite number of elements. The set which is not finite is known as the infinite set. The number of elements present in an infinite set is not finite and extends up to infinity.

Definition

A set which is empty or consists of a definite number of elements is called finite otherwise, the set is called infinite.

Examples for Finite Sets

A set of even natural numbers less than $11,A=\{2,4,6,8,10\}$ . Set $A$ has $5$ elements which is a finite number and the elements can be counted.

$B=\{$ Solution of the equation $x^{2}-16=0$ $\}$
$C=\{$ days of the week $\}$

Examples for Infinite Sets

we represent a set in the roster form, we write all the elements of the set within braces $\{\}$ . It is not possible to write all the elements of an infinite set within braces $\{\}$ because the numbers of elements of such a set is not finite. So, we represent some infinite set in the roster form by writing a few elements which clearly indicate the structure of the set followed (or preceded ) by three dots.

$\{1,2,3,...\}$ is the set of natural numbers

$\{1,3,5,7,...\}$ is the set of odd natural numbers

$\{2,4,6,8,...\}$ is the set of even natural numbers

$\{...,-3,-2,-1,0,1,2,3,...\}$ is the set of integers.

All these sets are infinite

Cardinality of Finite Set

If $a$ represents the number of elements of set $A$ , then the cardinality of a finite set is $n(A)=a$ . The cardinality of a finite set is a natural number or possibly $0$ .

So, the Cardinality of set $A$ of all English Alphabets is $26$ because the number of elements (alphabets) is $26$ .

Hence, $n(A)=26$ .

Similarly, a set containing the months in a year will have a cardinality of $12$ .

Hence, we can list all the elements of any finite set and list them in the curly braces or Roster form.

Cardinality of Infinite Sets

The cardinality of a set is $n(A)=x$ , where $x$ is the number of elements of a set $A$ . The cardinality of an infinite set is $n(A)=\infty$ as the number of elements is unlimited in it.

Properties of Finite Sets

A proper subset of a finite set is finite.
The union of any number of finite sets is finite.
The intersection of two finite sets is finite.
The cartesian product of finite sets is finite.
The cardinality of a finite set is a finite number and is equal to the number of elements in the set.
The power set of a finite set is finite.

Properties of Infinite Sets

The union of any number of infinite sets is an infinite set.
The power set of an infinite set is infinite.
The superset of an infinite set is also infinite.
A subset of an infinite set may or may not be infinite.
Infinite sets can be countable or uncountable. For example, the set of real numbers is uncountable whereas the set of integers is countable.