Finite Sets – Definition and Examples

Mathematics is incomplete without numbers. Hence, it is essential to develop a sound understanding of numbers. Sets could help us achieve that. The never-ending list of numbers in mathematics can be classified by using sets. 

In this section, we will be developing an understanding of Finite Sets.

In simpler words, finite sets are defined as:

Finite sets are the sets containing countable or finite numbers or elements. They are also called countable sets.

In this section of finite sets, we will be covering the following topics:

  • What is a finite set?
  • How to prove that a set is finite?
  • Properties of finite sets.
  • Examples
  • Practice Problems 

 

What is a Finite Set?

In real life, anything can be quantified as either being countable or uncountable. The countable items are classified as ‘finite,’ whereas the uncountable items are referred to as ‘infinite.’ A finite set consists of countable numbers. 

We can rephrase this statement by declaring that all the items or elements that can be counted are finite, while those items or elements that cannot be counted are infinite. Let’s take two examples: a basket of apples and the stars in the universe. In these examples, you can easily count the apples in the basket and but it is highly impossible to even count all the stars in the universe. Therefore, apples in the basket can be classified as finite, while the universe’s stars can be declared infinite. 

Mathematics is the universe of numbers. With unlimited numbers exceeding up to infinity, we need to learn to classify them as either finite or infinite to simplify the world around us. This classification can help distinguish finite from infinite and rational from irrational and can be achieved using sets.

In general terms, we can define a set as a group or a collection of numbers enclosed and contained in two brackets. When the contained items can easily be counted, the set will be classified as a finite set. 

Now, let’s see how we can notify a finite set. 

Notation of The Finite Set:

If ‘A’ represents a number system with a starting and an ending point, then all the elements in A can be counted and can be classified using a finite set.

The notation of finite sets is the same as that of any other set. Let’s consider the same number system A containing finite or countable elements. The numbers in this set, though they may be 100 or a billion, as long as they have an ending point, will be classified in a finite set. To open and close a finite set, curly brackets {} are used. The number system A can have the following notation:

A = {numbers in number system A} 

All the countable elements will be included in the finite set and will have the same notation as shown above. If we have more than one finite set in hand, we can notify each set independently by giving them a separate and distinguished notation. For example, using the above number system A, we can also denote this as following:

Number system = {numbers in number system A}

Or

X = {numbers in number system A}

So, you can use a phrase, a word, or even a letter to denote a finite set. 

Let’s consider some examples to comprehend the concept of the finite set further.

Example 1

P = {1,2,3,4,5,…..,10}

X = {x : x is an integer and 2<x<20}

Alphabets = {A,B,C,……..,Z}

Set of primary numbers till 10 = {2,3,5,7}

Example 2

Identify whether the following sets are finite or not:

(i)   Peach orchards in the country.

(ii)  People living in a town

(iii) People living in the world.  

Solution

We will solve this example by keeping in mind the concept of countable and uncountable. 

(i) The total number of peach orchards in the country can easily be counted, and yes, it can be classified as a finite set. The notation would be somewhat like the follows:

Peach Orchards = {no. of peach orchards in the country}

(ii) The total number of people living in a town can easily be counted and records. Hence, this can be classified into a finite set and can have the following notation:

Town People = {number of people living in town}

(iii) The total number of people living on earth cannot be counted as the number fluctuates with every passing second, and it is impossible to keep track of these numbers down to the last one. Hence, the world population cannot be classified as a finite set. 

How to Prove That a Set is Finite?

A set can only be considered a finite set if it contains countable items in it. To prove that a given set is a finite set, we will consider a number system. 

Mathematics itself is a huge realm consisting of numbers. But to prove that whether a given set is a finite set or not, we will consider the fundamental set of natural numbers. The set of natural numbers is a set that starts from 1 and has no limited end to it, just like numerical counting. In fact, it can last up to billions and even trillions. So to prove whether a set is a finite set or not, we will compare it with the set of natural numbers. 

Consider a set of natural numbers as given below:

N = {1,2,3,…………….,k}

Now, let’s consider a set A, which needs to be proved whether it is finite or not.

One simple trick to obtain the answer is to compare set A with set N. 

If set A actually lies in the set of natural numbers N, then the set can be declared as a finite set.

In mathematical terms, we can state this as:

N = {1,2,3,…………….,k}

A = {x,y,z,……………..,n}

If, x ϵ k and y ϵ k, and also x ϵ k

Or, n ϵ k

It can then be stated that set A actually belongs to the set of natural numbers N, and hence, set A is a finite set. 

Let’s solve some examples to understand this concept better.

Example 3

Prove that the set X = {4,5,8,12} is a finite set.

Solution

To prove that set X is a finite set, let’s consider the set of natural numbers, which is as follows:

N = {1,2,3,4,5,6,7,8,9,10,11,12,……….,n}

Now, let’s compare the two sets N and X, and let’s compare each element of X with the set of natural numbers N.

We can see the following results:

1st element of set X = 4 ϵ N

2nd element of set X = 5 ϵ N

3rd element of set X = 8 ϵ N

4th element of set X = 12 ϵ N

Since all the set X elements are actually natural numbers and have an ending point, the set X is a finite set.

Example 4

Check whether the set S = {x:x is a prime number and 2<x<17} is a finite set.

Solution

To check whether the set is a finite set or not, we will first convert it into a solvable set.

It is evident that the set S contains prime numbers and the range of these primary numbers is between 2 and 17. 

So, set S can be written as :

S = {3,5,7,11,13}

To check whether the set S is a finite set or not, we will compare its elements with the set of natural numbers N. 

N = {1,2,3,4,5,6,7,8,9,10,11,12,13,………….,k}

Now, let’s compare these elements.

1st elements of set S = 3 ϵ k

2nd element of set S = 5 ϵ k

3rd element of set S = 7 ϵ k

4th element of set S = 11 ϵ k

5th element of set S = 13 ϵ k

Since all these elements of set S actually belong to the set of natural numbers and have an ending point, set S can be stated as a finite set.

Properties of a Finite Set

A finite set is surely a unique set and contains countable and real items in it. These sets help us to classify and distinguish between countable items and uncountable items. Emphasizing the importance of finite sets and how they help simplify mathematics, we will consider some essential properties of finite sets to develop a thorough and deep understanding of finite sets.

1. Subset of Finite Set:

The subset of a finite set will always be a finite set. 

This concept can be understood by understanding the idea of subsets. A subset is basically a baby set that contains some of the elements of the parent set. Adhering to this statement, we can state that every finite set that contains natural numbers is actually a subset of the set of natural numbers. 

The subset of a finite set will always be a finite set, which can be understood with the following statements’ help.

Consider any finite set A that contains n finite elements. Since the set is a finite set, so it is bound to contain natural numbers. 

Now, consider a set a that is the subset of set A, and it contains (n-1) or (n-2) elements. Since this set a originates from set A, which contained natural numbers, set a will also have natural numbers. 

Hence, we can state that the subset set a of the set A is also a finite set.

Let’s consider this concept better with the help of examples.

Example 5

Consider a set S = {1,2,3,4} which is a finite set. Prove that the subset s = {1,2} is also a finite set.

Solution

Set S = {1,2,3,4} has 4 elements and all these elements are natural numbers.

Now, consider the subset s = {1,2}.

As the 1st element of s is a natural number and the 2nd element is also a natural number, the subset s is also a finite set.

2. Union of Finite Sets:

The union of two or more finite sets will always be a finite set.

Union of sets is actually defined as the joint junction of 2 or more sets. A union of 2 or more sets contains all the elements contained by the sets being unified.

The union of two or more finite sets will always be a finite set, which can be understood since the sets being unified are finite sets. Hence, they will contain natural numbers, so their joint set, which contains all the elements of the finite sets being unified, will also contain finite and natural numbers and hence will also be a finite set. 

We can understand this concept better with the help of an example.

Example 6

Consider 2 finite sets A = {1,3,5} and B = {2,4,6}. Prove that thier union is also a finite set. 

Solution

The two sets A and B are finite sets, and both contain natural numbers.

Their union can be expressed as:

A U B = {1,3,5} U {2,4,6}

A U B = Z = {1,2,3,4,5,6}

Now, the set Z, which indicates A and B’s union, contains the same elements from the finite sets, and these elements are all actually natural numbers. Hence, the union of sets A and B is also a finite set. 

3. Power Set of Finite Set:

The power set of a finite set is always a finite set. 

The power set of any set can be found by raising the power of 2 by the total number of elements in the finite set. 

To prove that the power set of a finite set is also a finite set, let’s consider the following example:

Example 7

Prove that the power set of the finite set S = {1,2,3,4} is also a finite set.

Solution

To find the power set, we have to calculate the number of elements in the set S.

As it is evident that set S has a total number of 4 elements, its power set can be found as:

The power set of S = 2^4

The power set of S = 16

As 16 is a natural number, the finite set’s powerset is also a finite set.

So that is all the information regarding finite sets required to enter the world of sets in mathematics. To further strengthen the understanding and the concept of a finite set, consider the following practice problems. 

Practice Problems 

  1. Check whether the following sets are finite sets:

        (i) A = {1,6,8,33456}          (ii) B = {x:x is an odd number and 3<x<∞}

  1. State whether the following sets are finite sets:

        (i)   Peach orchards of the world.

       (ii)  Hair on the human head.

       (iii) Chips in a Pringles box.

  1. Prove that the subset of the set A = {55,77,88,99} is a finite set.
  2. Prove that the union of the sets X = {2,4,6,8} and Y = {3,6,9,12} is a finite set.
  3. Prove that the power set of S = {10,20,30,40,50,60,70} is a finite set.

Answers

  1. (i) Finite    (ii) Not a finite set.
  2. (i) Finite    (ii) Not a finite set     (iii) Finite
  3. Finite
  4. Finite
  5. Finite 


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