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FINITE SETS AND INFINITE SETS

Finite sets and Infinite sets are totally different from each other. As the name
suggests, the finite set is countable and contains a finite number of elements.
The set which is not finite is known as the infinite set. The number of elements
present in an infinite set is not finite and extends up to infinity. Please note
that we can have countable infinite sets such as the set of rational numbers. We
come across various finite sets and infinite sets in our daily lives.

In this article, we will explore the concept of finite sets and infinite sets,
their definitions, and their properties. We will also understand the difference
between finite sets and infinite sets with the help of examples for a better
understanding.

1. What are Finite Sets? 2. Finite Sets Definition 3. What are Infinite Sets? 4.
Infinite Sets Definition 5. Difference Between Finite Sets and Infinite Sets 6.
Properties of Finite Sets 7. Properties of Infinite Sets 8. Finite Sets and
Infinite Sets Venn Diagram 9. FAQs on Finite Sets and Infinite Sets


WHAT ARE FINITE SETS?

Finite sets are sets having a finite or countable number of elements. It is also
known as countable sets as the elements present in them can be counted. In the
finite set, the process of counting elements comes to an end. Starting and
ending elements are present in the set. Finite sets can be easily represented in
roster notation form. For example, the set of vowels in English alphabets, Set A
= {a, e, i, o, u} is a finite set as the number of elements of the set are
finite.


FINITE SETS DEFINITION

Finite Sets are defined as sets with a finite number of elements. Elements of
finite sets can be counted. Please note that all finite sets are countable but
not all countable sets are finite. For example, consider a set of even natural
numbers less than 11, A = {2, 4, 6, 8, 10}. As we can see, set A has 5 elements
which is a finite number and the elements can be counted.


WHAT ARE INFINITE SETS?

Infinite sets can be understood as sets that are not finite. The elements of
infinite sets are endless, that is, infinite. If any set is endless from start
or end or both sides having continuity then we can say that set is infinite. For
example, the set of whole numbers, W = {0, 1, 2, 3, ……..} is an infinite set as
the number of elements is infinite. The set of real numbers is an example of
uncountable infinite sets. The elements of an infinite set are represented by
dots as the dots represent the infinity of the set.


INFINITE SETS DEFINITION

Infinite sets in set theory are defined as sets that are not finite. The number
of elements in an infinite set goes to infinity, that is, we cannot determine
the exact number of elements. Although we can have countable infinite sets whose
elements can be counted. For example, the set of integers, Z = {……… -2, -1, 0,
1, 2, ……….} is a countable infinite set as the number of elements in the set is
infinite and its elements can be put in one-to-one correspondence with the set
of natural numbers.


DIFFERENCE BETWEEN FINITE SETS AND INFINITE SETS

There are several similarities and differences between finite sets and infinite
sets. Some of the common differences are summarized in the table below:


FINITE SETS VS INFINITE SETS

Finite Sets Infinite Sets All finite sets are countable. Infinite sets can be
countable or uncountable. The union of two finite sets is finite. The union of
two infinite sets is infinite. A subset of a finite set is finite. A subset of
an infinite set may be finite or infinite. The power set of a finite set is
finite. The power set of an infinite is infinite. Example: Set of even natural
numbers less than 100, Set of names of months in a year Example: Set of points
on a line, Real numbers, etc.


PROPERTIES OF FINITE SETS

Now that we know the concept of finite sets, let us discuss some of its
properties:

 * A proper subset of a finite set is finite.
 * The union of any number of finite sets is finite.
 * The intersection of two finite sets is finite.
 * The cartesian product of finite sets is finite.
 * The cardinality of a finite set is a finite number and is equal to the number
   of elements in the set.
 * The power set of a finite set is finite.


PROPERTIES OF INFINITE SETS

Let us go through some of the important properties of infinite sets:

 * The union of any number of infinite sets is an infinite set.
 * The power set of an infinite set is infinite.
 * The superset of an infinite set is also infinite.
 * A subset of an infinite set may or may not be infinite.
 * Infinite sets can be countable or uncountable. For example, the set of real
   numbers is uncountable whereas the set of integers is countable.


FINITE SETS AND INFINITE SETS VENN DIAGRAM

A Venn diagram is formed by overlapping closed curves, mostly circles, each
representing a set, or in other words, it is a figure used to show the
relationships among sets, or groups of objects. The given below image of the
Venn diagram shows the relation between finite set and infinite set.



In the above image, Set containing elements {1, 13, 27} is a finite set, and a
set of natural numbers and a set of whole numbers are infinite sets. There are
multiple finite sets that can be created from an infinite set. The image given
above is showing one example of it where a finite set is lying inside infinite
sets.

Important Notes on Finite Sets and Infinite Sets

 * An empty set is a finite set with cardinality equal to zero.
 * The cardinality of rational numbers is equal to the cardinality of natural
   numbers.
 * All finite sets are countable whereas infinite sets may or may not be
   countable.

Related Topics

 * Operations on Sets
 * Universal Set
 * Venn Diagram
 * Intersection of Sets
 * Subsets

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FINITE SETS AND INFINITE SETS EXAMPLES

 1. Example 1: State whether the following sets are finite sets or infinite
    sets:
    
    a) Set A = Set of multiples of 10 less than 201
    
    b) Set of all integers.
    
    Solution: a) Set A = Set of multiples of 10 less than 201 = {10, 20, 30, 40,
    50,…., 200} is a finite set because the number of multiples of 10 less than
    201 is finite.
    b) Set of all integers is an infinite set because there is an infinite
    number of elements in the set.

 2. Example 2: Given, Set B = {x : x is an integer between -50 and 50}. Find out
    whether the given set is a finite or infinite set.
    
    Solution: Set B = {x: x is an integer between -50 and 50} is a finite set
    because the number of integers between -50 and 50 is finite.

 3. Example 3: Given, Set T = {….., -2, -1, 0}. Find out whether the given set
    is a finite or infinite set.
    
    Solution: Set T = {….., -2, -1, 0} is an infinite set because the elements
    of the set T start from negative of infinity and hence, cannot be finite.

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FINITE SETS AND INFINITE SETS QUESTIONS

 1. Q.1. Which of these is a finite set?
    Responses
     * {1, 2, 3, 4, 5}
       {1, 2, 3, 4, 5}
     * {1, 8, 9, 11, ...................}
       {1, 8, 9, 11, ...................}
    
    
    Check Answer
 2. Q.2. Which of these is an infinte set?
    Responses
     * Set of integers
       Set of integers
     * Set of months in a year
       Set of months in a year
    
    
    Check Answer

Check Answer >
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FAQS ON FINITE SETS AND INFINITE SETS


WHAT ARE FINITE SETS AND INFINITE SETS IN SET THEORY?

A set that has a finite number of elements is said to be a finite set, for
example, set D = {1, 2, 3, 4, 5, 6} is a finite set with 6 elements. If a set is
not finite, then it is an infinite set, for example, a set of all points in a
plane is an infinite set as there is no limit in the set.


WHAT IS THE DIFFERENCE BETWEEN FINITE SETS AND INFINITE SETS?

Finite sets are sets that have a fixed number of elements, are countable, and
can be written in roster form. An infinite set is a set that is not finite,
infinite sets may or may not be countable. This is the basic difference between
finite sets and infinite sets.


IS EMPTY SET A FINITE SET?

An empty set is a finite set as it contains no elements. The number of elements
in an empty set is definite, that is, zero, therefore, it is a finite set.


WHAT IS THE CARDINALITY OF A FINITE SET?

The cardinality of a finite set is the number of members or elements present in
the set. For example, set A is a set of all English alphabets, is a finite set.
The cardinality of the set of English alphabets is 26 as it contains 26
elements.


WHAT ARE THE EXAMPLES OF FINITE SETS AND INFINITE SETS?

Some common examples of finite sets and infinite sets are given below:

 * Let W be the set of the days of the week. Then W is a finite set.
 * Let Q be the set of points on a line. Then Q is an infinite set.
 * Let S be a set of stars in the sky, then S is an infinite set.
 * Let M be a set of months in a year, then M is a finite set.


HOW TO FIND THE CARDINALITY OF INFINITE SETS?

The cardinality of countable infinite sets is equal to the cardinality of
natural numbers, denoted by ℵ0 (which is the smallest infinite number). Whereas
the cardinality of uncountable infinite sets is either ℵ1 or greater than this.


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