What are Finite and Infinite Sets with Properties and Examples
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,...}.
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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, Vedantu maths experts have explained the properties of the sets.
Properties of Finite sets
Following are the conditions of the finite sets.
Two subsets always form a subset.
The power set of finite sets is finite.
A subset of finite sets is finite.
Properties of Infinite sets
Following are the conditions of the infinite sets.
Union of two infinite sets is an infinite set.
The power set of infinite sets is infinite.
The superset of the infinite set is an infinite set.
Graphical Representation of Finite and Infinite Sets
Also, Vedantu maths teachers have provided you with a graphical representation of the sets that clear the unwanted confusion about finite sets.
In addition to that, Vedantu 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?
The set of months of a year.
{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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FAQs on Finite and Infinite Sets in Set Theory
1. What is a finite set in mathematics?
A finite set is a set that contains a limited or countable number of elements. In other words, its elements can be counted and the counting process ends at a specific number.
- If a set A has n elements, we write n(A) = n.
- Example: A = {1, 2, 3, 4} has 4 elements, so it is finite.
- The empty set ∅ is also finite because it has 0 elements.
2. What is an infinite set?
An infinite set is a set that contains an unlimited number of elements and cannot be completely counted. The counting process never ends.
- Example: The set of natural numbers N = {1, 2, 3, ...} is infinite.
- The set of integers Z = {..., -2, -1, 0, 1, 2, ...} is also infinite.
- Infinite sets do not have a fixed cardinal number like finite sets.
3. What is the difference between a finite set and an infinite set?
The main difference between a finite set and an infinite set is that a finite set has a fixed number of elements, while an infinite set has endlessly many elements.
- Finite set: n(A) = a specific whole number (e.g., 5).
- Infinite set: Elements continue without end (e.g., natural numbers).
- Example: {2, 4, 6} is finite, but {2, 4, 6, 8, ...} is infinite.
4. How do you know if a set is finite or infinite?
A set is finite if you can count its elements and reach a final number, and infinite if counting never ends. To determine this:
- Step 1: Check if the elements stop at a specific value.
- Step 2: If there is a last element, the set is finite.
- Step 3: If the pattern continues forever (like “…”) it is infinite.
5. What is the cardinality of a finite set?
The cardinality of a finite set is the total number of elements it contains. It is denoted by n(A) or |A|.
- If A = {a, b, c}, then |A| = 3.
- The empty set has cardinality 0.
- Cardinality helps in solving problems involving subsets and counting.
6. Can you give examples of finite and infinite sets?
A finite set has limited elements, while an infinite set has unlimited elements.
- Finite set example: {red, blue, green} (3 elements).
- Finite number set: {10, 20, 30, 40}.
- Infinite set example: Natural numbers {1, 2, 3, ...}.
- Infinite number set: Even numbers {2, 4, 6, 8, ...}.
7. Is the empty set finite or infinite?
The empty set is a finite set because it contains 0 elements. Although it has no elements, zero is a definite number.
- It is denoted by ∅ or {}.
- Its cardinality is |∅| = 0.
- Since 0 is finite, the empty set is finite.
8. What are subsets of finite and infinite sets?
A subset is a set whose elements are all contained within another set. For a finite set with n elements, the number of subsets is 2n.
- If A = {1, 2}, then n = 2 and number of subsets = 2² = 4.
- Subsets: ∅, {1}, {2}, {1,2}.
- An infinite set has infinitely many subsets.
9. Are all number sets finite or infinite?
Most standard number sets such as natural numbers, integers, rational numbers, and real numbers are infinite sets. These sets extend without end.
- Natural numbers N = {1, 2, 3, ...} are infinite.
- Integers Z are infinite in both directions.
- A limited collection like {1, 2, 3, 4, 5} is finite.
10. What is a power set of a finite set?
The power set of a finite set is the set of all its subsets, and it contains 2n elements if the original set has n elements.
- If A = {a, b}, then n = 2.
- Power set P(A) = {∅, {a}, {b}, {a,b}}.
- Number of elements in P(A) = 2² = 4.
