Finite and Infinite Sets – Explanation, Properties and Solved Example

Definitions of Finite Set and Infinite Set

A finite set in mathematics is a set that has a finite number of elements. In simple words, it is a set that you can 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}.

An infinite set is a set with an uncountable number of elements. We use dots to represent the infinite elements in a roaster. For example, a set of infinite natural numbers. {1,2,3,4,...}.

Join Vedantu to read about these topics in detail from the expert maths faculties. 


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 an 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.

 

What will you learn in the Finite and Infinite Sets Chapter?

In this particular chapter, you come to know about the basic definitions of the finite set and infinite sets. Further in the topic, Vedantu maths experts have explained the properties of the sets. 

  • Properties of Finite sets

Following are the conditions of the finite sets.

  1. Two subsets always form a subset.

  2. The power set of finite sets is finite.

  3. A subset of finite sets is finite.

  • Properties of Infinite sets 

Following are the conditions of the infinite sets.

  1. Union of two infinite sets is an infinite set.

  2. The power set of infinite sets is infinite.

  3. The superset of the infinite set is an infinite set.

  • Graphical Representation of Finite and Infinite Sets

Also, Vedantu maths teachers have provided you with a graphical representation of the sets that clear the unwanted confusion about finite sets. 

In addition to that, Vedantu has brought a good amount of solved examples to help you understand these topics thoroughly and expose you to higher-order thinking skills (HOTS) questions.


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.

Answer: 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.


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FAQs (Frequently Asked Questions)

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

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.

2. Write the Properties of a Finite Set.

  • 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.

3. Why should you join Vedantu to read Finite and Infinite Sets?

In mathematics, the topic of finite and infinite sets is very important to read even Vedantu experts have confirmed this, and that’s why they have provided you notes with a detailed explanation of the topic. 

Vedantu offers you lots of resources to concrete your knowledge like notes of the important topics, previous papers. Also, for online tuition, we offer Vedantu pro classes for classes 1 to 12. All you need to do is register with your email id or phone number.

4. How can I download Vedantu notes of Finite sets and infinite sets?

Vedantu has offered you notes of all important topics of mathematics to help you ace your examinations. These notes will assist you in your school and the competitive examinations. The Finite and Infinite sets chapter will bring a positive response to your grades if you study it with Vedantu as experts maths teachers pen down these resources after analysing the previous questions and the syllabus of the Board. All these resources are free to download from the website, or you can install the Vedantu learning app to download it directly on your mobile device.

5. How to find the cardinality of the infinite sets?

When it comes to the cardinality of the finite set, we consider the number of the elements present in the set, but for the infinite set, counting is not possible. We no longer can speak of the number of elements present in a particular infinite set. You can find a concerning explanation of the topic in the notes; you can download it after signing up on the Vedantu. For a better picture of the topic, you can register for the Vedantu online class and get a chance to clear your doubts from the best maths teacher.

6. Does an empty set is a finite set?

Yes, an empty set is a finite set. An empty set does not contain any elements; blank curly brackets {} represent the empty set. You can easily count the number of elements in the finite set, and the elements inside the empty set are always zero, then the cardinality of the empty set is zero (0). If you have a value of the cardinality, it means the particular set has a definite number of elements, and this is what makes an empty set a finite set. 

7. What is the advantage of the Vedantu pro classes?

In the Vedantu pro class, you get a chance to learn from the best teachers of that particular subject, and you understand the concepts with3-D visual examples that break down the concepts more clearly. With Vedantu pro class, you can clear your doubts with experts one-on-one, and the most important thing is it is available to you on a screen, no need to go anywhere for different classes; everything is available at one touch of the screen.

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