Finite and Infinite Sets are two different kinds of Sets. Word Finite means countable and infinite stands for uncountable. Learn about Definite, Indefinite Sets Definition, Properties along with solved examples. To know more about this refer to the **Set Theory **and know more about different types of Sets.

### Finite Sets Definition

A Finite Set is a Set in which a number of elements are countable. In other words, Finite Sets are also called Countable Sets as they can be counted.

Example:

Set A has Months in a year i.e. { January, February, March, April, May, June, July, August, September, October, November, December}

Here n(A) = 12 countable elements thus it is a Finite Set.

Set B includes Vowels in English Alphabet i.e. { a, e, i, o, u}

n(B) = 5 countable elements thus it is a Finite Set.

**Cardinality of Finite Sets:** If a denotes the Cardinality of Finite Set A then n(A) = a. We can list out all the elements of a finite set and list them in curly braces.

### Properties of Finite Set

The following finite set conditions always hold true and are always finite.

- Union of Two Finite Sets.
- The subset of Finite Set
- Power Set of a Finite Set

Examples

P = {5, 12, 14, 20, 24, 30}

Q = {5, 10, 15, 20, 25, 30}

R = {5, 20, 30}

R ⊂ P, i.e. R is a Subset of P since all the elements of R are present in P. Thus, the Subset of Finite Set is Finite.

P, Q, R are finite sets since the number of elements in it is countable.

P U Q = { 5, 10, 12, 14, 15, 20, 24, 25, 30}

The power set of 2^{3} = 8, the number of elements of Set P is 3. This indicates that the power set of a finite set is finite.

### Non-Empty Finite Set

In this kind of Set, the number of elements in the set is quite large and only the beginning and ending of numbers are given. We can represent it as n(A) and if n(A) is a natural number then we consider it at some point as a Finite Set.

Consider M is a Set of Natural Numbers less than the number m. Thus we can say that the Cardinality of Set M is m.

### Infinite Sets Definition

If a Set is not finite or has uncountable elements then the Set is considered to be an Infinite Set. Since the number of elements is not countable it is also called Uncountable Set. We can’t represent Infinite Sets in Roster Form and the elements of Infinite Set are represented using …. denoting the infinity of the set.

Example

- Set of Points on the line
- Set of all Natural Numbers
- Set of all Integers.

**Cardinality of Infinite Sets: **

Cardinality of Set A is n(A) = x where x is the number of elements in the Set A. Cardinality of Infinite Set is n(A) = ∞ since the number of elements in it is unlimited.

### Properties of Infinite Sets

- The Superset of an infinite set is also infinite
- The Union of two infinite sets is infinite
- The Power set of an infinite set is infinite