# Finite Sets and Infinite Sets

What are the differences between finite sets and infinite sets?

Finite set: A set is said to be a finite set if it is either void set or the process of counting of elements surely comes to an end is called a finite set.

In a finite set the element can be listed if it has a limited i.e. countable by natural number 1, 2, 3, ……… and the process of listing terminates at a certain natural number N.

The number of distinct elements counted in a finite set S is denoted by n(S). The number of elements of a finite set A is called the order or cardinal number of a set A and is symbolically denoted by n(A).

Thus, if the set A be that of the English alphabets, then n(A) = 26: For, it contains 26 elements in it. Again if the set A be the vowels of the English alphabets i.e. A = {a, e, i, o, u} then n(A) = 5.

Note:

The element does not occur more than once in a set.

Infinite set: A set is said to be an infinite set whose elements cannot be listed if it has an unlimited (i.e. uncountable) by the natural number 1, 2, 3, 4, ………… n, for any natural number n is called a infinite set.

A set which is not finite is called an infinite set.

Now we will discuss about the examples of finite sets and infinite sets.

Examples of finite set:

1. Let P = {5, 10, 15, 20, 25, 30}

Then, P is a finite set and n(P) = 6.

2. Let Q = {natural numbers less than 25}

Then, Q is a finite set and n(P) = 24.

3. Let R = {whole numbers between 5 and 45}

Then, R is a finite set and n(R) = 38.

4. Let S = {x : x ∈ Z and x^2 – 81 = 0}

Then, S = {-9, 9} is a finite set and n(S) = 2.

5. The set of all persons in America is a finite set.

6. The set of all birds in California is a finite set.

Examples of infinite set:

1. Set of all points in a plane is an infinite set.

2. Set of all points in a line segment is an infinite set.

3. Set of all positive integers which is multiple of 3 is an infinite set.

4. W = {0, 1, 2, 3, ……..} i.e. set of all whole numbers is an infinite set.

5. N = {1, 2, 3, ……….} i.e. set of all natural numbers is an infinite set.

6. Z = {……… -2, -1, 0, 1, 2, ……….} i.e. set of all integers is an infinite set.

Thus, from the above discussions we know how to distinguish between the finite sets and infinite sets with examples.

Set Theory