Finite and Infinite Set: Class 11 Maths Notes, Definition, Examples, Properties & Solved Examples

A finite set contains a specific number of elements. These elements can be listed and counted. 

The process of enumeration concludes after a finite number of steps. Also, every finite set has a starting and an ending point.

A set is termed finite if its elements can be completely counted using natural numbers, and the process terminates.

Examples of Finite Sets

Set of vowels in the English language:

$\mathrm{V}=\{\mathrm{a},\mathrm{e},\mathrm{i},\mathrm{o},\mathrm{u}\}$
Set of prime numbers less than 20 :

$P=\{\mathrm{2,3},\mathrm{5,7},\mathrm{11,13,17,19}\}$

Set of colors in a rainbow:
$R=\{$  red, orange, yellow, green, blue, indigo, violet $\}$
In each of these cases, the number of elements can be precisely determined and the set listing ends.

Key Properties of Finite Sets

1. Every finite set has a well-defined cardinality (the number of elements).

2. The union, intersection, and cartesian product of finite sets are themselves finite.

3. Every subset of a finite set is finite.

4. The power set (set of all subsets) of a finite set is also finite.

Infinite sets, on the other hand, are collections that do not end. They continue without bound and cannot be counted entirely.

An infinite set is a set that is not finite. Its elements are unlimited in number and cannot be completely listed.

Examples of Infinite Sets

Set of natural numbers: $N=\{\mathrm{1,2},\mathrm{3,4},5,\dots \}$
Set of integers: $Z=\{\dots ,-3,-2,-\mathrm{1,0},\mathrm{1,2},3,\dots \}$
Set of real numbers between 0 and 1

Have a deeper understanding of finite and infinite sets is useful for JEE exams. 

• These are essential components of Sets in the JEE Main syllabus. 
• Knowing how you can differentiate finite and infinite sets would help you solve problems on set operations.
• This is a fairly easy and conceptual section in the exams, which makes it easier to score well.  

1. Infinite sets do not have a finite cardinality
2. An infinite set's subset could be finite, or it could be infinite, in its entirety.
3. Infinite sets are not only countable but also can be uncountable
4. Two infinite sets' union is generally infinite.
5. The power set of an infinite set is also infinite, and often has a higher order of infinity.

In Chapter 1 of your NCERT textbook, the definition of finite and infinite sets is

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

Basics of finite and infinite sets you should remember

1. When you can count the elements, it's a finite set
2. When the number is indefinite or uncountable, it's an infinite set

Finite Sets:

1. Elements are limited and countable.

2. Examples: $\left\{1,2,3\right\},\{a,bc\}$

3. Power set is finite.

4. Cardinality is a natural number.

5. Subsets are always finite.

6. Listing elements ends after finite steps.

Infinite Sets:

1. Elements are unlimited and not completely countable.

2. Examples: Set of natural numbers, set of points on a line

3. Power set is infinite.

4. Cardinality is infinite (countable or uncountable).

5. Subsets can be finite or infinite.

6. Listing elements never ends.

Imagine a circle representing a finite set, like $\{1,13,27\}$ . Now, picture a larger space representing the infinite set of all natural numbers and set of whole numbers.