Infinite Set

An infinite set is a set with infinite elements, i.e., the number of elements in an infinite set never depletes, regardless of the finite elements we remove.

Mathematically, A set of elements A is said to be infinite if the elements of its proper subset A' can be put into one-to-one correspondence with the elements of A.

If an infinite set can be put into one-to-one correspondence with the set of natural numbers, then the infinite set is called a countable infinite set; otherwise, it is called an uncountable infinite set.

Infinities can be of different sizes or levels, for example, a set of Real Numbers (R) is a larger infinity than a set of Natural or Rational Numbers, but both the set of natural numbers and rational numbers are the same size sets, as both have the same cardinality.

Infinite Set Notation

The infinite sets are denoted by using dots (. . .) at the end of the elements, which follow a pattern. Any set A that is an infinite set can be represented as follows:

A = {a₁ , a₂ , a₃ , . . . }

Let A be an infinite set, then it is denoted as A = {1, 2, 3,...}., here we can see that 1, 2, 3,... is a common pattern that leads to further elements such as 4, 5, 6, and so on. Thus, there are infinitely many elements of set A, which can simply be represented by {1, 2, 3, . . .}.

Infinite Set Examples

Some examples of infinite sets are:

• Natural Numbers: The set of natural numbers (N) represents the counting numbers, starting from 1 and going on infinitely (1, 2, 3, 4, ...).
• Prime Numbers: The set of prime numbers is an infinite set that contains numbers greater than 1 that have no divisors other than 1 and themselves (e.g., 2, 3, 5, 7, 11, 13, ...).
• Integers: The set of integers (Z) includes all positive and negative whole numbers and zero, i.e., { . . . -3, -2, -1, 0, 1, 2, 3, . . .}.
• Rational Numbers: The set of rational numbers (Q) includes all numbers that can be expressed as a fraction of two integers, such as 1/2, -3/4, 5/7, etc.
• Irrational Numbers: The set of irrational numbers consists of numbers that cannot be expressed as a fraction of two integers, such as √2, π, and e.
• Real Numbers: The set of real numbers (R) contains all numbers, both rational and irrational, such as 3.14, √2, π, e, etc.
• Cantor Set: Cantor Set is a mathematical construct created by removing the middle third of a line segment repeatedly, forming an infinite set of points. It is self-replicating and has a surprising property of being uncountably infinite yet having zero length on the number line.

Types of Infinite Sets

Infinite Sets are classified into two categories that are:

Countable Infinite Set

A set X is called a countable infinite set if and only if set A has the same cardinality as N (natural numbers). Some examples of countable infinite sets are a set of natural numbers N, a set of integers Z, etc.

Uncountable Infinite Set

A set that can't be mapped to the set of natural numbers is called the Uncountable Infinite Set. We can also say that an uncountable infinite set is a set that is not countable. The set of real numbers R is one of the examples of an uncountable infinite set.

Cardinality of an Infinite Set

The number of elements in the set is called the cardinality of the set. For example, if A = {1, 3, 4} then |A| = 3. Historically, the concept of cardinality was only associated with finite sets, as this seemed logical at the time. But a man with a vision to revolutionize the understanding the infinity came, and the name of the man is Georg Cantor. He revolutionized the concept of the infinite in the context of sets and their cardinalities. He devised the use of ℵ to present the cardinality of infinite sets.

Cardinality of countably infinite set = ℵ₀

ℵ₀ = |N| = |Q| = |Z|

Cardinality of lowest uncountably infinite set = ℵ₁ OR C (Continuum)

C OR ℵ₁ = |R| = |P(N)|

Cardinality of an uncountably infinite set = ℵ₂

ℵ₂ = |P(P(N))|

Continuum Hypothesis

The Continuum Hypothesis states that there is no set whose cardinality is strictly between that of the set of natural numbers (denoted by ℵ₀) and the set of real numbers (denoted by C i.e., Continuum). This was proposed by the great mathematician Georg Cantor in 1878 as part of his groundbreaking work on the theory of infinite sets.

The Continuum Hypothesis can be stated as follows:

• ℵ₀ = |N| (cardinality of the natural numbers)
• C = |R| (cardinality of the real numbers)

The Continuum Hypothesis claims that there is no set A such that ℵ₀ < |A| < C.

Properties of Infinite Sets

Various properties of Infinite sets are discussed below.

• Any subset of a countable set is countable.
• A superset of an uncountable set is uncountable.
• If A is an infinite set, it can be either countably infinite or uncountably infinite.
• The union of two or more countable sets is countable.
• The Cartesian product of two countable sets is countable.
• If A is an uncountable infinite set and B is any set, then the Cartesian product of A and B is also an uncountable infinite set.

How to Check If a Set is Infinite?

To check whether a set is finite or infinite, we use the following methods:

For Infinite Sets

Observe Patterns: If any set follows a specific pattern for its elements without any end, it should be an infinite set.

• For example, the set of natural numbers {1, 2, 3, 4, ...} is infinite, as you can see the pattern of adding 1 to each further element without any end.

Bijection with a known infinite set: If you can establish a one-to-one and onto correspondence (bijection) between the given set and a known infinite set, then the given set is also infinite.

• For example, we can establish a bijection between a set of integers and natural numbers, and we know that the set of natural numbers is an infinite set. Thus, a set of integers is also an infinite set.

Finite Set vs Infinite Set

Related Articles

• Representation of Sets
• Union of Sets
• Types of Sets
• Intersection of Sets

Solved Examples on Infinite Sets

Example 1: Check whether the given sets are finite or infinite.

(i) A = {3, 6, 9, ...}

(ii) B = {1, 2}

(iii) X = {a, b, c, ...}

Answer:

(i) A = {3, 6, 9, ...}

A is an infinite set.

(ii) B = {1, 2}

The number of elements in B is 2 which is finite. So, B is a finite set.

(iii) X = {a, b, c, ...}

X is an infinite set.

Example 2: Check whether the given sets are finite or infinite.

(i) Z = {15, 19, 21, 45}

(ii) P = {"infinite", "set", ...}

(iii) Q = {..., p, q, r, s, ...}

Answer:

(i) Z = {15, 19, 21, 45}

The number of elements in Z is 4 which is finite. So, Z is a finite set.

(ii) P = {"infinite", "set", ...}

P is an infinite set as it does not have end point.

(iii) Q = {..., p, q, r, s, ...}

Q is an infinite set as it does not have start and end point.

Example 3: Check whether the given sets are finite or infinite.

(i) Factors of 25

(ii) Multiples of 2

(iii) Line segments in a plane

Solution:

(i) Factors of 25

There are only 3 factors of 25 ( 1, 5 and 25). So, the set of factors of 25 is finite set.

(ii) Multiples of 2

Multiples of 2 are {2, 4, 6, 8, ...}

There is no end point for the set of multiples of 2. So, the set multiples of 2 is infinite set.

(iii) Lines segments in a plane

There can be uncountable line segments in a plane. So, the set of line segments in a plane is an infinite set.

Example 4: Prove the power set of set A = {2, 3, 5, ...} is an infinite set.

Solution:

A = {2, 3, 5, ...} is an infinite set.

From the property of infinite set, we know that power set of infinite set is infinite.

So, power set of A is an infinite set.

Practice Question on Infinite Sets

Question 1: Check whether the given sets are finite or infinite.
(i) M = {2, 4, 6, 8, ...}
(ii) N = {1, 3, 5, 7, 9}
(iii) W = {apple, banana, cherry, ...}

Question 2: Check whether the given sets are finite or infinite.
(i) Prime numbers less than 100
(ii) Days of the week
(iii) Rational numbers between 0 and 1

Question 3: Check whether the given sets are finite or infinite.
(i) Set of even numbers less than 10
(ii) Set of all integers
(iii) Set of prime numbers greater than 1

Question 4: Prove that the power set of set B = {1, 2, 3, ...} is an infinite set.

Question 5: Prove that the power set of a finite set is always finite.