Finite and Infinite Sets: Definition, Examples, Calculator, Questions

Q: Define finite and infinite sets

A finite set is a set that has a small number of elements which can be counted by infinite sets are uncountable.

Q: Give some examples for finite and infinte sets.

Finite and infinite sets examples are, finite sets - set of all prime numbers less than $10$, set of all even numbers less than $1000$, set of all alphabets, etc. Infinite sets - set of all integers, set of all numbers between $1$ and $2$, set of all perfect squares, etc.

Q: Distinguish between finite and infinite sets.

In finite sets, elements are countable but in infinite sets, elements are not countable.

Q: What are the criteria to identify whether two sets are of finite or infinite magnitude?

A set is said to be finite when it is possible for the set to contain an end number of elements or it is countable. For instance, the alphabet in English is well defined since it has 26 elements of letters including upper case and lower case letters. If there is no bound to what you are able to count in the set then the set is said to be an infinity. For instance, let us take the set of natural numbers, which is an infinite set that extends all the way up into the infinity without termination.

Q: What is the difference between a finite set and an infinite set?

A finite set contains a countable number of elements that can be listed completely, while an infinite set has an endless number of elements that cannot be fully listed. For example, {1, 2, 3, 4, 5} is finite, but the set of all natural numbers is infinite.

Q: How can you determine if a set is finite or infinite?

To determine if a set is finite or infinite, consider whether you can count all its elements. If you can list all elements or express the total number of elements as a natural number, it's finite. If the elements go on endlessly without a final count, it's infinite.

Q: Can an infinite set have a cardinality?

Yes, infinite sets have cardinality, but it's not represented by a finite number. Instead, we use cardinal numbers like ℵ₀ (aleph-null) for countably infinite sets, or larger cardinals for uncountably infinite sets. This allows us to compare the "sizes" of different infinite sets.

Q: Can an empty set be considered finite or infinite?

An empty set, denoted as {} or ∅, is considered finite. It contains zero elements, which is a definite, countable number. This might seem counterintuitive, but in set theory, the empty set is treated as a finite set with cardinality zero.

Q: What is a countably infinite set?

A countably infinite set is an infinite set whose elements can be put into a one-to-one correspondence with the natural numbers. In other words, its elements can be "counted" using natural numbers, even though the counting never ends. The set of all integers is an example of a countably infinite set.

Q: What is the cardinality of a finite set?

The cardinality of a finite set is the number of elements in the set. It's always a non-negative integer. For example, the cardinality of {a, b, c} is 3, while the cardinality of an empty set is 0.

Finite and Infinite sets

Edited By Komal Miglani | Updated on Jul 02, 2025 06:38 PM IST

Consider the set of all even numbers less than 100, the elements in this set is countable. Now consider the set of all natural numbers, the elements in this set is not countable as the natural numbers go on till infinity. A set with countable elements is a finite set whereas a set in which the number of elements is uncountable is infinite set.

This Story also Contains

What are Finite and Infinite Sets?
Finite set
Infinite Set
Distinguish between Finite and Infinite Sets
Solved Examples Based On the Finite and Infinite sets

Finite and Infinite sets

A finite set contains countable quantities of something, for example, how many students exist in a single class. The uncountable set can be defined as a set that has an infinity and, therefore, the number of members in the set cannot be counted like the natural numbers set that has no finite upper limit. In this article, let's look into finite and infinite sets definition and examples for finite and infinite sets

This article is about the concepts of the finite set and infinite set. This concept falls under the broader category of sets relation and function, a crucial Chapter in class 11 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more.

What are Finite and Infinite Sets?

Sets are simply a collection of distinct objects, considered as a whole. These objects, called elements or members of the set, can be anything: numbers, people, letters, etc. Sets are particularly useful in defining and working with groups of objects that share common properties.

It is a well-defined collection of distinct objects and it is usually denoted by capital letters A, B, C, S, U, V…...

Finite set

A set that is empty or consists of a finite number of elements is called a finite set.

Examples: $φ, {a}, {1, 2, 5, 9}, {x : x$ is a person of age more than 18 $}$

Properties of finite Sets

A subset of the Finite set is finite
The union of two or more finite sets is finite
The power set of a finite set is countable

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Cardinality of Finite Set

If ' $a$ ' represents the number of elements of set $A$ , then the cardinality of a finite set is $n (A) = a$ . The cardinality of a finite set is a natural number or possibly $0$ , it can be either.

So, the Cardinality of set $A$ of all English Alphabets is 26 because the number of elements (alphabets) is $26$ .

Hence, $n (A) = 26$ .

Infinite Set

A set which has infinite elements is called an infinite set.

Examples: A set of all the lines passing through a point, set of all circles in a plane, set of all points in a plane, $N, Z, Q, Q^{'}, R$ , ${x : 2 < x < 2.1}$

Properties of Infinite Sets

The union of two or more infinite sets is infinite
The power set of an uncountable infinite set is infinite
The superset of an infinite set yeilds an infinite set

Cardinality of Infinite Sets

The cardinality of a set is $n (A) = x$ , where $x$ is the number of elements of a set $A$ . The cardinality of an infinite set is $n (A) = \infty$ as the number of elements is unlimited in it.

Distinguish between Finite and 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.

Note: Empty and singleton sets are finite sets.

Solved Examples Based On the Finite and Infinite sets

Example 1: Which of the following sets is a finite set?

1) $P = {$ natural numbers greater than $50}$
2) $Q = {$ integers less than $5}$
3) $R = {$ whole numbers more than $10}$
4) $S = {$ natural numbers less than $5}$

Solution

There are only $4$ natural numbers less than $5$ . So it is a finite set.

Hence, the answer is the option 4.

Example 2: Which of the following is an example of an infinite set?

1) Set of all the persons living in India.

2) Set of all the human beings living on Mars.

3) Set of all the stars in the Universe.

4) Set of the satellites of Earth.

Solution:

As there are infinite stars in the Universe, so it is an infinite set.

Note that the population of India is finite.

Hence, the answer is the option 3.

Example 3: Which of the following is not an infinite set?

1) Set of all real numbers.

2) Set of all perfect squares.

3) Set of all the divisors of $x$ , where $x ∈ N$.

4) Set of all prime numbers.

Solution:

The number of divisors of a number is finite.

Hence, the answer is the option 3.

Example 4: Classify the following as finite and infinite sets?

1) Set of all natural numbers which are neither a prime nor a composite number.

2) Set of even prime numbers.

3) Set of numbers that divide $12$ and $20$ .

4) $S = {ϕ}$ .

Solution:
(1) set $= {1}$ - finite set
(2) set $= {2, 3, 5, 7, 13, . . . . . .}$ - infinite set
(3) Set - ${2, 4}$ - finite set
(4) ${ϕ}$ - finite set

Example 5: Which of the following sets is a finite set?
1) Set of all points in a plane.
2) Set of all points on a line segment.
3) Set of all lines in a plane.
4) Set of all circles passing through three non-collinear points.

Solution

In this question,

Number of geometrical points and lines in a plane are infinite. Also, the number of points in a line segment is infinite.

However, there is only one circle passing through three non-collinear points, so it is a finite set (as it has one element).

Hence, the answer is the option 4.

List of Topics Related to Finite and Infinite Sets

Roster And Set Builder Form Of Sets	Universal Set
Empty Set	Union Of Sets, Properties Of Union
Subsets And Types Of Subsets	Intersection Of Set, Properties Of Intersection
Equal And Equivalent Sets	Difference Of Set
Singleton Set	Complement Of A Set, Law Of Complement, Property Of Complement
Power Set	De-morgan's Laws

Frequently Asked Questions (FAQs)

1. Define finite and infinite sets

A finite set is a set that has a small number of elements which can be counted by infinite sets are uncountable.

2. Give some examples for finite and infinte sets.

Finite and infinite sets examples are,

finite sets - set of all prime numbers less than $10$ , set of all even numbers less than $1000$ , set of all alphabets, etc.

Infinite sets - set of all integers, set of all numbers between $1$ and $2$ , set of all perfect squares, etc.

3. Distinguish between finite and infinite sets.

In finite sets, elements are countable but in infinite sets, elements are not countable.

4. What are the criteria to identify whether two sets are of finite or infinite magnitude?

A set is said to be finite when it is possible for the set to contain an end number of elements or it is countable. For instance, the alphabet in English is well defined since it has 26 elements of letters including upper case and lower case letters. If there is no bound to what you are able to count in the set then the set is said to be an infinity. For instance, let us take the set of natural numbers, which is an infinite set that extends all the way up into the infinity without termination.

5. Is

{0}

finite or infinite set ?

Finite set.

6. What is the difference between a finite set and an infinite set?

A finite set contains a countable number of elements that can be listed completely, while an infinite set has an endless number of elements that cannot be fully listed. For example, {1, 2, 3, 4, 5} is finite, but the set of all natural numbers is infinite.

7. How can you determine if a set is finite or infinite?

To determine if a set is finite or infinite, consider whether you can count all its elements. If you can list all elements or express the total number of elements as a natural number, it's finite. If the elements go on endlessly without a final count, it's infinite.

8. Can an infinite set have a cardinality?

Yes, infinite sets have cardinality, but it's not represented by a finite number. Instead, we use cardinal numbers like ℵ₀ (aleph-null) for countably infinite sets, or larger cardinals for uncountably infinite sets. This allows us to compare the "sizes" of different infinite sets.

9. Can an empty set be considered finite or infinite?

An empty set, denoted as {} or ∅, is considered finite. It contains zero elements, which is a definite, countable number. This might seem counterintuitive, but in set theory, the empty set is treated as a finite set with cardinality zero.

10. What is a countably infinite set?

A countably infinite set is an infinite set whose elements can be put into a one-to-one correspondence with the natural numbers. In other words, its elements can be "counted" using natural numbers, even though the counting never ends. The set of all integers is an example of a countably infinite set.

11. What is the cardinality of a finite set?

The cardinality of a finite set is the number of elements in the set. It's always a non-negative integer. For example, the cardinality of {a, b, c} is 3, while the cardinality of an empty set is 0.

12. What is the power set of a finite set?

The power set of a set is the set of all possible subsets, including the empty set and the set itself. For a finite set with n elements, its power set contains 2^n elements. For example, the power set of {1, 2} is {∅, {1}, {2}, {1, 2}}, which has 2^2 = 4 elements.

13. Can a set be both finite and infinite?

No, a set cannot be both finite and infinite. These are mutually exclusive properties. A set is either finite (has a countable number of elements) or infinite (has an endless number of elements), but never both simultaneously.

14. What is the union of a finite set and an infinite set?

The union of a finite set and an infinite set is always an infinite set. This is because the resulting set includes all elements from both sets, and the presence of infinitely many elements from the infinite set makes the union infinite, regardless of the finite set's size.

15. What is the Cartesian product of a finite set and an infinite set?

The Cartesian product of a finite set A and an infinite set B is an infinite set. It contains all ordered pairs (a, b) where a is from set A and b is from set B. Since B is infinite, there are infinitely many such pairs for each element of A, making the product infinite.

16. Are all infinite sets the same "size"?

No, not all infinite sets are the same "size". Some infinite sets are larger than others. For example, the set of real numbers is uncountably infinite and larger than the set of natural numbers, which is countably infinite. This concept is formalized in set theory using different levels of infinity.

17. Is the power set of a finite set always finite?

Yes, the power set of a finite set is always finite. If a set has n elements, its power set will have 2^n elements, which is a finite number. However, the power set of an infinite set is always infinite and, in fact, has a larger cardinality than the original set.

18. How does the concept of "countability" relate to finite and infinite sets?

Countability is a way to classify infinite sets. Finite sets are always countable, as their elements can be counted and listed. Infinite sets can be either countably infinite (like the set of natural numbers) or uncountably infinite (like the set of real numbers). Countably infinite sets can be put in a one-to-one correspondence with the natural numbers.

19. Can a subset of an infinite set be finite?

Yes, an infinite set can have finite subsets. For example, the set of natural numbers is infinite, but {1, 2, 3} is a finite subset of it. In fact, every infinite set has infinitely many finite subsets.

20. Can the intersection of two infinite sets be finite?

Yes, the intersection of two infinite sets can be finite, or even empty. For example, the intersection of the set of even numbers and the set of odd numbers is empty, even though both original sets are infinite. The intersection depends on the specific elements in each set, not just their size.

21. Can an infinite set be a subset of a finite set?

No, an infinite set cannot be a subset of a finite set. By definition, an infinite set contains an endless number of elements, while a finite set contains a limited number. Therefore, it's impossible for all elements of an infinite set to be contained within a finite set.

22. How does the concept of infinity relate to the real number line?

The real number line is an example of an uncountably infinite set. It contains infinitely many points, and between any two points, there are infinitely many more points. This "continuous" nature of the real number line makes it uncountably infinite, unlike the set of integers which is countably infinite.

23. How does the concept of infinity relate to the decimal representation of numbers?

The decimal representation of numbers can be finite (like 3.14) or infinite (like π = 3.14159...). Numbers with infinite decimal representations that don't repeat are uncountably infinite, while repeating decimals are countably infinite. This connects the concept of infinity to the way we represent numbers.

24. How does the concept of infinity relate to fractals?

Fractals are geometric shapes that exhibit self-similarity at increasingly small scales, potentially infinitely. While any physical representation of a fractal is finite, the mathematical concept of a fractal involves infinite detail. This connects the idea of infinity to geometric patterns and shapes.

25. How does the concept of infinity relate to the idea of limits at infinity in calculus?

In calculus, limits at infinity describe the behavior of functions as the input grows arbitrarily large. This is different from set-theoretic infinity, which describes set sizes. A function

26. What is the significance of Cantor's diagonal argument?

Cantor's diagonal argument is a proof technique that demonstrates the existence of uncountably infinite sets. It shows that the set of real numbers is "larger" than the set of natural numbers, proving that not all infinite sets have the same size. This was a groundbreaking result in set theory and our understanding of infinity.

27. What is the relationship between infinity and the concept of limits in calculus?

In calculus, infinity is often used to describe the behavior of functions as their input grows without bound. This is different from the set-theoretic notion of infinity, which describes the size of sets. In limits, we're often interested in how a function behaves as it approaches, but never reaches, infinity.

28. What is the relationship between infinite sets and the Axiom of Choice?

The Axiom of Choice is a fundamental principle in set theory that states that for any collection of non-empty sets, it's possible to select one element from each set to form a new set. This axiom is particularly important when dealing with infinite collections of sets and has far-reaching consequences in mathematics.

29. How does the concept of infinity relate to the precision of measurements in science?

In science, the concept of infinity often relates to the theoretical precision of measurements. While we can always imagine measuring something more precisely, in practice, there are limits to measurement precision due to factors like quantum uncertainty. This creates a distinction between mathematical infinity and practical limitations in scientific measurements.

30. What is the relationship between infinite sets and the concept of dimension in geometry?

In geometry, dimension is often related to the "size" of certain infinite sets. For example, a line (1D) contains uncountably many points, as does a plane (2D) or a cube (3D). However, higher-dimensional spaces can be shown to contain "more" points in a set-theoretic sense, connecting the concepts of dimension and different sizes of infinity.

31. How does the concept of infinity relate to the idea of convergence in series?

In the study of series, infinity appears in two ways: the series itself may have infinitely many terms, and the sum of the series may approach infinity. However, an infinite series can converge to a finite sum (like the geometric series with |r| < 1), showing that an infinite process can result in a finite outcome.

32. Can an infinite set contain only even numbers?

Yes, an infinite set can contain only even numbers. The set of all even natural numbers {2, 4, 6, 8, ...} is an example of such an infinite set. It's countably infinite and can be put in one-to-one correspondence with the set of all natural numbers.

33. What is the difference between a bounded infinite set and an unbounded infinite set?

A bounded infinite set has elements that are all within a finite range, while an unbounded infinite set has elements that can grow arbitrarily large. For example, the set of all numbers between 0 and 1 is a bounded infinite set, while the set of all natural numbers is an unbounded infinite set.

34. How does the concept of infinity relate to limits in calculus?

In calculus, infinity is often used to describe limits that grow without bound. While this is related to infinite sets, it's a slightly different concept. In set theory, infinity describes the size of a set, while in calculus, it often represents a process that continues without end or a value that grows arbitrarily large.

35. What is the cardinality of the set of all subsets of a finite set?

If a finite set has n elements, the cardinality of its power set (the set of all subsets) is 2^n. This is because for each element, we have two choices: include it in a subset or not. These choices lead to 2^n possible combinations, which is the number of subsets.

36. Can two infinite sets have different cardinalities?

Yes, two infinite sets can have different cardinalities. For example, the set of natural numbers and the set of real numbers are both infinite, but the set of real numbers has a larger cardinality. This demonstrates that there are different "sizes" of infinity.

37. What is the relationship between rational numbers and infinity?

The set of rational numbers is countably infinite, meaning it has the same cardinality as the natural numbers. This might seem counterintuitive since there are infinitely many rationals between any two integers, but they can still be put in a one-to-one correspondence with the natural numbers.

38. How does the concept of density relate to infinite sets?

Density in set theory refers to the property that between any two elements of a set, there is always another element of that set. The rational numbers and real numbers are both dense sets. This property often leads to infinite sets, but not all infinite sets are dense.

39. Can an infinite set have a largest or smallest element?

An infinite set can have a smallest or largest element, but it doesn't necessarily have to. For example, the set of natural numbers has a smallest element (1) but no largest element. The set of negative integers has a largest element (-1) but no smallest element. Some infinite sets, like the set of all integers, have neither.

40. What is the difference between a countably infinite set and an uncountably infinite set?

A countably infinite set can be put into a one-to-one correspondence with the natural numbers, meaning its elements can be "counted" using natural numbers (even though this counting never ends). An uncountably infinite set is "larger" and cannot be put into such a correspondence. The set of integers is countably infinite, while the set of real numbers is uncountably infinite.

41. How does the concept of infinity relate to geometric series?

In geometric series, infinity can appear in two ways. First, the series itself can be infinite if it continues forever. Second, the sum of an infinite geometric series can be finite (if |r| < 1) or infinite (if |r| ≥ 1). This demonstrates that an infinite process (adding terms forever) can sometimes result in a finite outcome.

42. What is the difference between potential infinity and actual infinity?

Potential infinity refers to a process that can continue without end, like counting natural numbers. Actual infinity, on the other hand, refers to a completed infinite totality, like the set of all natural numbers. The distinction is philosophical and has been debated throughout the history of mathematics.

43. Can an infinite set be divided into finite subsets?

Yes, an infinite set can be divided into finite subsets. For example, the set of natural numbers can be divided into finite subsets of consecutive numbers: {1, 2, 3}, {4, 5, 6}, and so on. This division results in infinitely many finite subsets.

44. What is the Continuum Hypothesis and how does it relate to infinite sets?

The Continuum Hypothesis states that there is no set with cardinality between that of the integers and the real numbers. It relates to the idea that there might be different sizes of infinity. Interestingly, this hypothesis can neither be proved nor disproved using standard set theory axioms.

45. Can an infinite set be closed?

Yes, an infinite set can be closed. In topology, a set is closed if it contains all of its limit points. For example, the set of all real numbers is both infinite and closed. The set of all integers is also infinite and closed in the real number line.

46. Can two finite sets have the same cardinality but different elements?

Yes, two finite sets can have the same cardinality (number of elements) but contain different elements. For example, {1, 2, 3} and {a, b, c} both have a cardinality of 3, but contain entirely different elements.

47. What is the significance of Hilbert's Hotel in understanding infinite sets?

Hilbert's Hotel is a thought experiment that illustrates some counterintuitive properties of infinite sets. It describes a hotel with infinitely many rooms, all occupied, yet still able to accommodate new guests. This paradox helps to demonstrate how infinite sets can be put into one-to-one correspondence with proper subsets of themselves.

48. How does the concept of cardinality extend to infinite sets?

For infinite sets, cardinality is extended using transfinite numbers. The smallest infinite cardinal number is ℵ₀ (aleph-null), which is the cardinality of the set of natural numbers. Larger infinite cardinals exist, such as the cardinality of the set of real numbers, denoted by 2^ℵ₀ or ℵ₁ (assuming the Continuum Hypothesis).

49. Can an infinite set have a finite complement?

Yes, an infinite set can have a finite complement, depending on the universal set. For example, if the universal set is the set of all integers, the complement of the infinite set of all positive integers is the finite set of non-positive integers {0, -1, -2, -3, ...}.

50. What is the difference between countable and uncountable sets?

Countable sets are those that can be put into a one-to-one correspondence with the natural numbers or a subset of the natural numbers. This includes all finite sets and some infinite sets like the integers or rational numbers. Uncountable sets, like the real numbers, cannot be put into such a correspondence and are "larger" than countable sets.

51. Can an infinite set be well-ordered?

Yes, an infinite set can be well-ordered. A well-ordered set is one in which every non-empty subset has a least element. The natural numbers are a classic example of an infinite well-ordered set. The Well-Ordering Theorem, which is equivalent to the Axiom of Choice, states that every set can be well-ordered.

52. Can an infinite set be bounded?

Yes, an infinite set can be bounded. For example, the set of all real numbers between 0 and 1 is infinite (uncountably so) but is bounded by 0 and 1. This shows that the property of being bounded is independent of whether a set is finite or infinite.

53. What is the relationship between infinite sets and the concept of density in number systems?

Density in number systems refers to the property that between any two numbers, there's always another number of the same type. Both rational and real numbers are dense, and both are infinite sets. However, the real numbers are uncountably infinite, while the rationals are only countably infinite, showing that density alone doesn't determine the "size" of infinity.