Finite and Infinite Sets

A set can be either finite or infinite based on the number of elements present in it. In this article, we will discuss their definitions, cardinalities, types, properties, and visual representations using Venn diagrams.

Finite Set

A set is considered finite if it contains a countable number of elements. It is thus also called a countable set.

Examples

A set of natural numbers less than 10 is Z = {1, 2, 3, 4, 5, 6, 7, 8, 9}, is finite.

Here are a few more examples of finite sets: 

• Set of all rainbow colors; A = {Red, Orange, Yellow, Green, Blue, Indigo, Violet}
• Set of vowels in the English alphabet;  D = {a, e, i, o, u}
• Set of all days in a week; B = {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}

In Roster Form

A finite set is represented in roster form by listing all its elements within curly braces {}. The elements are separated by commas. The general form of a finite set A in roster form is written as: 

A = {a₁, a₂, a₃, a₄….a_n}

here, 

• a₁, a₂, a₃, a₄….a_n are the elements of the finite set
• n is the total number of elements

With Venn Diagram

A finite set can be easily represented in a Venn diagram. 

Set of all rainbow colors A = {Red, Orange, Yellow, Green, Blue, Indigo, Violet}

The set of vowels in the English alphabet D = {a, e, i, o, u}

Set of days in a week B = {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}

Cardinality

If A is a finite set and x is the number of elements in set A, then the cardinality of set A; |A| or n(A) = x, which is either a positive integer or 0.

For example, the cardinality of a set of all days in a week 

B = {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday} is 

|B| or n(B) = 7,as it has 7 elements.

Similarly, the set containing the colors in the rainbow has a cardinality of 7. Thus, |A| or n(A) = 7

Properties

• The subsets of a finite set are finite.
• The union of any number of finite sets gives a finite set.
• The intersection of any two finite sets is finite.
• The cartesian products of finite sets are finite.
• Finite sets have a finite cardinality equal to the number of elements in the set.
• The power set of a finite set is finite.

Whether an Empty Set is Finite

Since an empty set has no elements, represented as {}, it is countable.

Thus, an empty set is finite.

Infinite Set

A set is infinite if it contains an uncountable number of elements. 

Examples

An example of an infinite set is the set of all natural numbers, X = {1, 2, 3, 4, 5, …}

A few more examples of infinite sets are:

• Set of natural numbers; ℕ = {1, 2, 3, …}
• Set of whole numbers; W = {0, 1, 2, …}
• Set of integers; ℤ = {…, -2, -1, 0, 1, 2, …}
• Set of real numbers; ℝ = {x: x ∈ ℚ ∪ ℚ’}

In Roster Form

An infinite set, in general, is written as:

A = {a₁, a₂, a₃, a₄…}

here, 

• a₁, a₂, a₃, a₄  are the first 4 elements of the infinite set

With Venn Diagram

It is difficult to represent an infinite set using a Venn diagram. Here is a Venn diagram showing the number system.

Types

Infinite sets are of two types:

Countable 

A countable infinite set is a set that has elements that can be arranged in a one-to-one correspondence with the set of natural numbers.

The set of natural numbers (ℕ), integers (ℤ), and rational numbers (ℚ) are a few examples of countable sets.

Uncountable 

An uncountable set is a set that has a cardinality larger than that of the set of natural numbers. Thus, their elements cannot be listed in a one-to-one correspondence with natural numbers. 

One example of such a set is the set of real numbers (ℝ).

Cardinality

Georg Cantor introduced infinite sets and defined their sizes. He used the symbol ‘ℵ’ to represent the cardinality of these sets. Here, ℵ is the first letter from the Hebrew language, also known as ‘aleph null’ (ℵ₀), representing the smallest infinite number.

The cardinality of a countably infinite set is ℵ₀, where ℵ₀ = |ℕ| = |ℚ| = |ℤ|

The cardinality of the lowest uncountably infinite set is ℵ₁ or C (Continuum), where C or ℵ₁ = |ℝ| = |P(ℕ)|

The continuum hypothesis deals with the possible sizes of infinite sets. It states that there is no such set whose cardinality is strictly between the set of natural numbers (ℵ₀) and the set of real numbers (C). 

Mathematically, it states that there is no such set, say set A, such that ℵ₀ < |A| < C. Thus, the cardinality of an uncountably infinite set is ℵ₂ and ℵ₂ = |P(P(ℕ))|

Properties

• Any subset of an infinite set is infinite.
• The union of any number of infinite sets results in an infinite set.
• A superset of an infinite set is infinite.
• A power set of an infinite set yields an infinite set.
• The cartesian product of two countably infinite sets is a countable infinite set. Also, the cartesian product of any set and an uncountable infinite set results in an uncountable infinite set.

Finite vs. Infinite Sets 

Here are the differences between finite and infinite sets.

Factors	Finite Sets	Infinite Sets
Number of elements	They are countable.	They are either countable or uncountable.
Bounded/Unbounded	They have the start and end elements. Thus, they are bounded.	They are endless from the start/end or both sides. Thus, they are unbounded.
Union	The union of two sets is finite.	The union of two infinite sets is infinite.
Subset	A subset of a finite set is finite.	A subset of an infinite set is either finite or infinite.
Power set	The power set of a finite set is finite.	The power set of an infinite set is infinite.
Example	The set of the first 7 natural numbers.	The set of natural numbers.

Venn Diagram

The Venn diagram below shows the relation between a finite and an infinite set:

Here, the set A = {1, 2, 3} has 3 elements, thus a finite set. Whereas the set ℕ = {1, 2, 3, 4, …} has infinite elements, it is thus an infinite set.

Solved Examples