Finite Sets

Finite set is a collection of finite, well-defined elements. For better understanding, imagine you have a bunch of your favourite toys or snacks. You know exactly how many you have, that’s the idea of a finite set in math. A finite set is a way to discuss collections of things you can count. In this article, we will discuss the concept of finite sets and their properties. We will also understand the difference between finite sets and infinite sets with the help of examples for a better understanding.

What is Finite Set?

Finite sets are sets with a finite or countable number of elements. These sets are also known as countable sets because their elements can be counted. The process of counting elements comes to an end in a finite set. The set consists of an element from the beginning and an element from the end. Finite sets are easily represented in roster notation. An example of a finite set is the set of vowels in the English alphabet, Set A = [a, e, i, o, u], which has a finite number of elements.

Finite Sets Definition

Finite Sets are sets that contain a finite number of elements or the elements of finite sets can be counted.

Consider the set A = [a, e, i, o, u]; elements can be counted in this set, so it can be considered a finite set.

Note: All finite sets are countable, but not all countable sets are finite.

Venn Diagram of Finite Set

As Venn Diagrams are used to Finite Sets, lets consider set of all vowels A i.e., A = { a, e, i, o, u} and a universal set i.e., set of all english alphabets. The relation between both these sets can be represented using the following diagram.

Learn more about Venn Diagram.

Example of Finite Set

There are various examples of finite sets, some of those examples are as follows:

Set of Colors in a Rainbow	{Red, Orange, Yellow, Green, Blue, Indigo, Violet}
Set of Planets in the Solar System	{Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune}
Set of Days in a Week	{Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}
Set of Playing Cards on one Suit in a Deck	{Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King}
Set all Outcomes of throwing a Dice	{1, 2, 3, 4, 5, 6}
Set of Letters in the English Alphabet	{A, B, C, . . . , X, Y, Z}

Cardinality of Finite Set

The number of elements in a finite set is known as the cardinality of that set. It represents the size or count of elements within the set. The cardinality of any finite set A is represented by the symbol n(A).

Let’s consider some examples of Cardinality, as follows:

• Set A = {1, 2, 3, 4, 5}: Cardinality of set A is n(A) = 5
• Set B = {apple, banana, orange, pear}: Cardinality of set B is n(B) = 4
• Set C = {cat, dog, bird}: Cardinality of set C is n(C)= 3

Properties of Finite Sets

The following finite set conditions are always finite.

• Finite sets have a definite count of elements (cardinality).
• Every subset of a finite set is also finite.
• The union and intersection of finite sets are finite.
• Empty set is a finite set with no elements.
• Power set of a finite set is also finite.
• Cartesian product of finite sets is finite.

Finite and Infinite Sets

Let’s compare the differences between the Finite and Infinite sets in the following table:

Finite Set	Infinite Set
Finite sets are countable.	Infinite sets can be countable or uncountable.
A subset of a finite set will always be finite.	A subset of an infinite set may be finite or infinite.
The power set of a finite set will always be finite.	The power set of an infinite set is infinite.
The union of two finite sets is finite.	The union of two infinite sets is infinite.
Example: Set of odd natural numbers less than 50,  Set of names of days in a week	Example: Set of points on a line, Set of Real numbers, etc.

Solved Problems on Finite Sets

Problem 1: Let A = {1, 2, 3, 4} and B = {3, 4, 5, 6} be finite sets. Find the union and intersection of sets A and B.

Solution:

• The union of two sets, denoted by A ∪ B, is the set of all elements that are in A, or in B, or in both.
  • A ∪ B = {1, 2, 3, 4, 5, 6} (Combining all unique elements from both sets)
• The intersection of two sets, denoted by A ∩ B, is the set of all elements that are common to both A and B.
  • A ∩ B = {3, 4} (Only the elements 3 and 4 are present in both sets)

Problem 2: Let C = {2, 4, 6} be a finite set. Find the power set of set C.

Solution:

The power set of a set is the set of all possible subsets, including the empty set and the set itself.

Power set of {2, 4, 6} = {{}, {2}, {4}, {6}, {2, 4}, {2, 6}, {4, 6}, {2, 4, 6}}

The power set of the set {2, 4, 6} contains 2³ = 8 subsets, including the empty set and the set itself.

Problem 3: Let A = {1, 2, 3, 4, 5} be a finite set. How many subsets does set D have?

Solution:

To find the number of subsets of a finite set, we can use the formula 2ⁿ, where n is the number of elements in the set.

In this case, A has 5 elements, so the number of subsets = 2⁵ = 32.

Set A has 32 subsets, including the empty set and the set itself.

Problem 4: Let A = {2, 4, 6, 8, 10} and B = {1, 3, 5, 7, 9} be two finite sets. Find the cardinality of the union of sets A and B.

Solution:

The union of two sets A and B is the set containing all the elements that are in A, B, or both. So, the union of A and B is {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. The cardinality of this set is the number of elements it contains, which is 10. Hence, the cardinality of the union of sets A and B is 10.

Problem 5: Let C = {1, 2, 3, 4, 5} and D = {3, 4, 5, 6, 7} be two finite sets. Find the cardinality of the intersection of sets C and D.

Solution:

The intersection of two sets C and D is the set containing all the elements that are common to both C and D. So, the intersection of C and D is {3, 4, 5}. The cardinality of this set is the number of elements it contains, which is 3. Hence, the cardinality of the intersection of sets C and D is 3.

Problem 6: You have two finite sets that have m and n elements”, In how many different ways can we create a new set by combining elements from both given sets?

Solution:

When combining elements from two sets, we can think of each element as a choice. For each element in Set A, we have n choices from Set B to pair it with. Since there are m elements in Set A, and each element has n choices from Set B, the total number of ways to combine them is m * n.

Problem 7: Which of the following is a finite set?

1. Set of all prime numbers
2. Set of odd numbers between 1 and 100
3. Set of real numbers between -1 and 1

Solution:

As there are infinitly many prime number, and also between any two real numbers  there are infinitely many numbers, Thus, only finite set between all three is only the 2. As there are only 50 odd numbers between 1 and 100.

Thus, the correct answer is 2. 

Practice Problems on Finite Sets

Problem 1: Let A = {2, 4, 6, 8} and B = {4, 6, 8, 10}. Find:

a) A ∪ B (Union of sets A and B)

b) A ∩ B (Intersection of sets A and B)

c) A’ (Complement of set A)

d) B’ (Complement of set B)

Problem 2: Given the universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and sets P = {2, 4, 6, 8, 10} and Q = {3, 6, 9}, find:

a) P ∩ Q’ (Elements in P but not in Q)

b) (P ∪ Q)’ (Complement of the union of sets P and Q)

c) U’ (Complement of the universal set)

Problem 3: Let A, B, and C be finite sets. Prove or disprove: If A ⊆ B and B ⊆ C, then A ⊆ C.

Problem 4: Let A, B, and C be finite sets. Prove the distributive law for sets: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).

FAQs on Finite Sets

What is Finite Set in Set Theory?

A finite set is a collection of distinct elements that has a definite and countable number of members. It can be listed, and its elements can be counted.

How do you Determine the Cardinality of a Finite Set?

The cardinality of a finite set is the number of elements it contains. To find the cardinality, count the elements in the set.

Are all Subsets of a Finite Set also Finite?

Yes, every subset of a finite set is also finite. The number of subsets of a finite set can be determined using the formula 2^n, where n is the cardinality of the original set.

Define Finite Set.

A finite set is like a group of things that we can count. It is like having a box of toys where w ecan count how many toys are inside the box because there’s a limited number of them.

How is a Finite Set different from an Infinite Set?

A finite set has a countable and limited number of elements, while an infinite set has an generally uncountable number of elements, which could be potentially endless.

How do you Represent a Finite Set?

A finite set can be represented using curly braces { } that enclose its elements. For example, the set of even numbers less than 10 can be represented as {2, 4, 6, 8}.

Can a Finite Set be empty?

Yes, a finite set can be empty. As empty set contain no elements and considered finite.

How do you Find the Size of a Finite Set?

The size of a finite set, also known as its cardinality and cardinality is the number of elements it contains. For example, the size of the set {apple, orange, banana} is 3.

What is the Power Set of a Finite Set?

The power set of a finite set is the set of all possible subsets of that set, including the empty set and the set itself. The power set is always larger than the original set.

Can a Finite Set have Duplicate Elements?

No, not only finite set but any set do not contain any duplicate element.