**Finite and Infinite set**

In previous articles we have learnt about Definition of Set, Methods of Representation of Set and Definition of Empty Set, In this article we are going to learn about Definition of Finite and Infinite Set.

In this article we are going to learn and develop a depth understanding about definition of finite and infinite set, their representation and their properties with some important examples

A finite and infinite sets are totally different from each other.

Anything in nature can be quantified as countable or uncountable. The quantity that is countable can be classified as finite and the quantity that is uncountable can be classified as infinite

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**Definition of Finite Set**

- In Mathematics a finite set is a set having finite numbers of elements.
- A finite set is also called as countable set because we can count and finish counting of elements in the set.
- The start and end elements are present in the finite sets therefore there is no continuity

** ****Examples of finite Set **

** ****(1)** Set of first 10 natural numbers can be represented as A= {1,2,3,4,5,6,7,8,9,10}

Here set A is a finite set and elements are countable, we have total 10 elements in set A first element is 1 and last element in 10.

**(2)** P = {x: x is integer and 99<x<201) Here Set P can be represented as

P = {100, 101, 102, 103, 104, 105 ….200}

Set P is a finite set and elements are countable, we have total 101 elements in set P, first element is 100 and last element is 200.

**(3)** Q = {x: x is all mathematics books in a school library}

In a school library the number of mathematics books are finite and countable so set Q is a finite set.

**(4)** R = {x: x is day of a week}, Here Set R can be represented as

R = {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}

Set R is a finite set because element in the set are countable.

**Representation of Finite Set **

The representation of finite sets is the same as that of any other set.

A finite set can be represented in two ways

**(1)** Roster form or Tabular form of representation

**(2)** Set builder form of representation

In Roster form of representation, a finite set can be represented as

A = {2,4,6,8,10} and P = {1,2,3,4,5……..99}

In Set builder form of representation, a finite set can be represented as

R = {x: x is a natural number and 10<x<20} and Q = {x: x is a prime number and x<1000}

**Properties of Finite Set**

Important properties of finite set are listed below

**The union of two finite sets **

The union of two finite sets is a finite set,

Let we have two finite sets P and Q Union of Sets P and Q is P U Q and this will produce a finite set

**Example**

Let P = {a, b, c, d, e, f} and Q = {d, e, f, g, h, j} so we can calculate Union of Set P and Q as

P U Q = {a, b, c, d, e, f, g, h, j} and this will be a finite set having total number of elements are fine and that is 9. So, union of two finite set is finite in nature.

**Cardinality of Finite Set**

The **cardinality** of a finite set is a measure of a set’s size. the Cardinality of a finite set P can be defined as the number of distinct elements counted in a finite set P and symbolically denoted by n(P) or |P|.

**Examples**

**(1)** Calculate the cardinality of Set A = {1,2,3,4,5,6}, Total number of elements in Set A = 6 So, Cardinality of Set A = n(A) = |A| = 6.

**(2)** Calculate the Cardinality of Set B = { }, Set B is the Empty set and total number of elements in Set B = 0 So, Cardinality of Set B = n(B) = |B| = 0.

**(3)** Calculate the cardinality of Set C = {1,2,3,4,5 ……..200} Total number of elements in Set C is 20

So, Cardinality of Set C = n(C) = |C| = 200

**Cardinality of the Union of two finite Sets **

Let we have two finite sets X and Y Union of Sets X and Y is X U Y and this will produce a finite set and cardinality of X U Y will be | X U Y | and described as | X U Y | = | X | + | Y | – | X ∩ Y |

**Example **

Let Set X = {1,2,3,4,5} and Y = {4,5,6,7}. So, cardinality of Set X = |X| = 5 and cardinality of Set Y = |Y| =4. and Intersection of set X and set Y = X ∩ Y = {4,5}

Cardinality of X ∩ Y = | X ∩ Y | = 2 and | X U Y | = 5+4-2 = 7, that is a finite number

**Power set of Finite Set**

For a given finite set A, the Power Set P(A) is defined as a set of subsets of finite set A and the number of elements in **Power set** P(A) are defined as 2^n(A), here n(A) is the cardinality of set A

In other words, Subsets of finite Set A are the elements of Power set P(A)

Number of elements in Power Set P(A) = 2^n(A)

**Example**

Let Set A = {0,1,2}

So, cardinality of set A = n(A) = 3

So, number of elements in power set P(A) = 2^3 =8

And subsets of finite set A are {0}, {1}, {2}, {0,1}, {0,2}, {1,2}, {0,1,2}, ϕ

Now, P(A) = { ϕ, {0}, {1}, {2}, {0,1}, {0,2}, {1,2}, {0,1,2} }

**Definition of Infinite Set**

- In Mathematics an infinite set is a set having infinite numbers of elements.
- An infinite set is also called as uncountable set because we cannot count and finish counting of elements in the set.
- The start element can present but end element is not present in the infinite sets therefore there is continuity

**Example of infinite Set **

Let a Set P = {x: x is natural number and x>5}, Here set P is an infinite set because there are infinite number of natural number greater than 5

Let a set R = {x: x = n^3 and n ϵ N}

Here set R = {0,1,8, 81 ………} is an infinite set because there are infinite number of elements in the set R

**Representation of Infinite Set **

The representation of infinite sets is the same as that of any other set. An infinite set can be represented in Set builder form but not in Roster form or Tabular form of representation.

In Set builder form of representation, an infinite set can be represented as

R = {x: x is a natural number and x>100}

**Properties of Infinite Set**

Properties of infinite set are listed below

**The union of two infinite sets **

The union of two infinite sets is an infinite set,

Let we have two infinite sets A and B Union of Sets A and B is A U B and this will produce an infinite set

**Example**

Let A = {x: x is parallel straight line and 2<x<4} and B = {x: x is number of planets in the universe} so we can calculate Union of Set A and B and this will produce a set with infinite number of elements, and A U B will be an infinite set.

**The union of an infinite set with a finite set**

The union of an Infinite set with a finite set gives an infinite set

**Example **

Let an infinite set P = {x: x is an integer and x>3} and a finite set Q = {1,2,3}, Union of Set P and Q will be P U Q = {x: x is a positive integer} will be an infinite set

**Cardinality of an infinite Set**

Th cardinality of an infinite set is infinite in nature

Let Set G = {9, 99, 999, 9999, 99999, ……},in given set G there is an infinite number of element so cardinality of Set G will be infinite

**Power set of an infinite Set **

Power Set P(H) of an infinite set H will be infinite in nature because for an infinite set there is an infinite number of subsets

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