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 orĀ theĀ 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
Letās 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:Ā
- Set of odd natural numbers less than 50,Ā
- Set of names of days in a week
| Example:Ā
- Set of points on a line,
- Set of Real numbers, etc.
|
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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 āŖ B, is the set of all elements that are in A, or in B, or in both.
- A āŖ B = {1, 2, 3, 4, 5, 6} (Combining all unique elements from both sets)
Ā
- The intersection of two sets, denoted by A ā© B, is the set of all elements that are common to both A and B.
- A ā© 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?
- Set of all prime numbers
- Set of odd numbers between 1 and 100
- Set of real numbers between -1 and 1
Solution:
As there are infinitly many prime number, and also between any two real numbers Ā 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.Ā
Practice Problems on Finite Sets
Problem 1: Let A = {2, 4, 6, 8} and B = {4, 6, 8, 10}. Find:
a) A āŖ B (Union of sets A and B)
b) A ā© 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 ā© Q' (Elements in P but not in Q)
b) (P āŖ 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 ā B and B ā C, then A ā C.
Problem 4: Let A, B, and C be finite sets. Prove the distributive law for sets: A ā© (B āŖ C) = (A ā© B) āŖ (A ā© C).
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