[Maths Class Notes] on Finite and Infinite Sets Pdf for Exam

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 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, maths experts have explained the properties of the 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.

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

In addition to that, 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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