Finite and Infinite Sets

In mathematics, sets are fundamental concepts used to group objects, numbers, or elements that share certain properties. These elements can be anything—numbers, letters, shapes, or even abstract entities. Sets are classified based on the number of elements they contain. Two primary types of sets are finite sets and infinite sets. 

1.0What are Finite Sets?

A finite set is a set that contains a specific, countable number of elements. These sets are limited in size, meaning they do not go on forever. You can easily list the elements of a finite set, and the process will come to an end after a fixed number of terms.

Examples of Finite Sets:

1. Set of natural numbers less than 10:

A = {1, 2, 3, 4, 5, 6, 7, 8, 9}

This set has exactly 9 elements, so it's a finite set.

2. Set of colors in a traffic light:

B = {Red, Yellow, Green} 

The set contains only 3 elements, making it a finite set.

3. Set of even numbers between 1 and 20:

C = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20}

This set contains 10 elements, and thus it is finite.

In all these cases, you can easily count the number of elements, and they don’t go on indefinitely.

2.0What are Infinite Sets?

An infinite set, on the other hand, is a set that contains an unlimited number of elements. These sets cannot be counted because they go on forever in at least one direction. Infinite sets can be further divided into countably infinite sets and uncountably infinite sets, depending on whether their elements can be listed in a sequence.

Examples of Infinite Sets:

1. Set of natural numbers:

N = {1, 2, 3, 4, 5, 6, 7, …}

The set of natural numbers goes on forever, so it is an infinite set.

2. Set of all points on a line:

L = {All points between 0 and 1} 

The set of points on a line segment, or even a line, is infinite because there are infinitely many points between any two numbers.

3. Set of all prime numbers:

P = {2, 3, 5, 7, 11, 13, …}

Prime numbers continue without end, so the set of all prime numbers is infinite.

3.0Key Differences Between Finite and Infinite Sets

1. Number of Elements:

• A finite set has a definite, countable number of elements.
• An infinite set has an uncountable or unlimited number of elements.

2. Cardinality:

• The cardinality of a finite set refers to the number of elements in the set and is a natural number.
• The cardinality of an infinite set cannot be expressed as a natural number—it is considered "infinite."

Examples:

• Examples of finite sets include the set of days in a week, set of months in a year, and set of students in a class.
• Examples of infinite sets include the set of integers, set of real numbers, and set of points on a circle.

Notation:

• Finite sets are typically denoted with a finite list of elements enclosed in curly brackets.
• Infinite sets are represented using ellipses (\dots) or set-builder notation, indicating that the set continues indefinitely.

4.0Solved Examples on Finite and Infinite Sets

Example 1:

Let A = {2, 4, 6, 8, 10}. Determine whether A is finite or infinite and find its cardinality.

Solution:

• The set A = {2, 4, 6, 8, 10} consists of 5 distinct elements.
• The number of elements in A is 5, which is a finite number.
• Therefore, A is a finite set.

Cardinality of the set A:

The cardinality of a set is the number of elements in the set. Here, the cardinality of A is 5, since it contains 5 elements.

Answer:

• A is a finite set.
• Cardinality of A = 5.

Example 2:

Consider the set of all natural numbers, N = {1, 2, 3, 4, 5, …}. Determine if N is finite or infinite.

Solution:

• The set N contains all natural numbers starting from 1, continuing indefinitely.
• Since the set goes on forever and the elements cannot be listed completely, N is an infinite set.

Cardinality of the set N:

• The cardinality of the natural numbers is not a finite number—it is countably infinite, meaning the elements can be listed in an infinite sequence (1, 2, 3, 4, 5, ...).

Answer:

• N is an infinite set.
• The cardinality of N is countably infinite.

Example 3:

Let B = { }. Determine if BB is finite or infinite, and find its cardinality.

Solution:

• The set B = { } is the empty set, which contains no elements.
• Since it has 0 elements, it is considered a finite set.

Cardinality of the set B: The cardinality of the empty set is 0 because it contains no elements.

Answer:

• B is a finite set.
• Cardinality of B = 0.

Example 4:

Consider the set of real numbers between 0 and 1, R = [0, 1]. Determine if R is finite or infinite.

Solution:

• The set R = [0, 1] contains all real numbers between 0 and 1, including decimals and fractions.
• Since the real numbers in this range are uncountably infinite (there is no way to list them all in a sequence), R is an infinite set.

Cardinality of the set R: The set of real numbers between 0 and 1 is uncountably infinite. Its cardinality is greater than the cardinality of the natural numbers and cannot be counted.

Answer:

• R is an infinite set.
• The cardinality of R is uncountably infinite.

Example 5:

Let C = {1, 2, 3, 4, 5}. Determine whether C is finite or infinite, and find its cardinality.

Solution:

• The set C = {1, 2, 3, 4, 5} contains exactly 5 elements.
• Therefore, it is a finite set.

Cardinality of the set C: The number of elements in C is 5, so the cardinality of C is 5.

Answer:

• C is a finite set.
• Cardinality of C = 5.

Example 6:

Let D = {1, 3, 5, 7, 9, …}. Determine if D is finite or infinite, and explain why.

Solution:

• The set D = {1, 3, 5, 7, 9, …} consists of all odd numbers starting from 1 and continuing indefinitely.
• As this set goes on forever, it is an infinite set.

Cardinality of the set D: The set of odd numbers is countably infinite, meaning you can list the elements (1, 3, 5, 7, 9, ...), but there are infinitely many of them.

Answer:

• D is an infinite set.
• The cardinality of D is countably infinite.