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Finite and Infinite Sets

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What is a Finite Set?

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In the set theory of mathematics, a finite set is defined as a set that has a finite number of elements. In other words, a finite set is a set which you could in principle count and finish counting. For example, {1,3,5,7} is a finite set with four elements. The element in the finite set is a natural number, i.e. non-negative integer. A set S is called finite if there exists a bijection f:S = {1,……,n} for natural number n. The empty set {} is also considered finite. So, S is a finite set, if S admits a bijection to some set of natural numbers of the form {|x| < n}.


The Cardinality of Finite Set

The cardinality of a finite set is n(A) = a, here a represents the number of elements of set A.

Whereas, the cardinality of the set A of all English Alphabets is 26, as the number of elements (alphabets) is 26. So, n(A) = 26.

It shows that you can list all the elements of a finite set and write them in curly braces or the form of Roster. Sometimes, the number of factors may be too big, but somehow it is countable or has a starting and ending point. Then this type of set is called a Non-Empty Finite Set. The number of elements is denoted with n(A) and if n(A) is a natural number then only it is a finite set.


Is Empty Set a Finite Set?

An empty set is a set which has no elements in it. It is represented as { }, which shows that there is no element in the given set. The cardinality of an empty set is 0 (zero) as the number of elements is zero.

A={ } or n(A)=0.

The finite set is a set with countable elements. As the empty set has zero elements in it, so it has a definite number of elements.

Therefore, an empty set is a finite set with cardinality zero.


What is the Infinite Set?

A set which is not a finite set is infinite. If the number of elements is uncountable, then also it is called an infinite set. Unlike finite sets, we cannot represent an infinite set in roster form easily as its elements are not limited. So, dots are used to describe the infinity of the set.


Properties of an Infinite Set

  • The union of two infinite sets is always infinite.

  • The power set of an infinite set is infinite.

  • The superset of an infinite set is also infinite.

Infinite Set Examples

  • Countably Infinite Set:

The set of all integers is countably infinite, even when it is a proper subset of the integers. The set of all countably rational numbers is also a countably infinite set as a bijection is present to the set of the integers.

  • Uncountably Infinite Set:

The set of all the real numbers is uncountably infinite. Also, the set of irrational numbers is uncountably infinite.


Graphical Representation of Finite and Infinite Sets

(image will be uploaded soon)


Here,

A= {1,2,3,4,5}.

B= {1,2,6,7,8}.

AUB= {1,2,3,4,5,6,7,8}.

A∩B= {1,2}.

Both A and B are finite sets as they have a limited number of elements so, AUB and A∩B are also finite.


Solved Example

Q1. Which of the following sets are finite or infinite?

  1. The set of months of a year.

  2. {1,3,5,…..}.

Answer: 1. The set of months can be represented as A= {Jan, Feb, Mar, Apr, May, Jul, Aug, Sep, Oct, Nov, Dec}. It forms a set of countable elements with the number of elements =12. Hence, it is a finite set.

2. The set {1,3,5,…} has all the natural numbers but does not consist of any ending point. This makes it an uncountable set, and so it is an infinite set.


Q2. What is the Cardinality of Infinite Sets?

Answer: Cardinality of a set is expressed as n(A) = x, where x is the number of elements in the set A.

The number of elements in an infinite set is unlimited, so the cardinality of the infinite set n(A) = infinity.

FAQ (Frequently Asked Questions)

Q1. How to Find Out if a Set is Finite or Infinite?

Answer: As we already know, that a set is finite if it has a starting and an ending point and a set are infinite if it has no end from either of the two sides. To determine if the set is finite or infinite, you should remember the following points:

  • The set having a starting and ending point is a finite set, but if it does not have a starting or ending point, it is an infinite set. If the set has a limited number of elements, then it is finite whereas if it has an unlimited number of elements, it is infinite.

  • If the cardinality of the set is n(A) = n it is finite, but if the cardinality of the set n(A) = infinity then it is an infinite set.

Q2. Write the Properties of a Finite Set.

Answer: 

  • The subset of a finite set is always finite.

  • The union of two finite sets is also finite.

  • The power set of a finite set is finite.

Example: A= {2,3,5,7}

B = {2,4,6,8,10}

C = {2,3}

Here all the sets are finite as the number of elements are limited and countable.

  • C is the subset of A, as all the elements of C are present in set A. So the subset of a finite set is finite always.

  • AB is {2,3,4,5,6,7,8,10}. This proves the union of two finite sets is also a finite set.

  • The number of elements of a power set is 2n. So, for the power set of A is 24= 16, as there are four elements in the set A. Hence, the power of a finite set is finite.