Cardinality of a set

In this article, we are going to discuss the cardinality of a set along with examples. Here, we will also discuss the cardinality of finite and infinite sets.

Hope, this article will be helpful to you in order to understand the cardinality of a set.

A set is a collection of distinct objects of the same type. Sometimes we are required to know the size of sets. Cardinality of a set is defined as the total number of unique elements in a set. As an instance, the set A = {a, b, c} has a cardinality of 3 as it contains only three elements.

It is easy to get the size of finite sets as they are well behaved. The problem occurs with the infinite set as they are difficult to understand.

Vertical bars are used to represent the cardinality of a set. For example, the cardinality of set A is denoted as ∣A∣. When A is finite, ∣A∣ will be the number of elements in the set A. When A is infinite, ∣A∣ is represented by a cardinal number. The cardinal number of a set is characterized as n (A), where A is any set and n(A) is the number of elements in the set A.

Note: The cardinality of an empty set is equal to zero, i.e., |∅|= 0.

The cardinality of the countably infinite set is countably infinite.

Now, let's discuss the cardinality of finite and infinite sets.

Cardinality of Finite sets

Before understanding the cardinality of finite sets, first, understand finite sets. A set is said to be finite if it contains exactly n distinct elements where n is a non-negative integer. Finite Sets are also called numerable sets. Here, n is said to be the "cardinality of sets." The cardinality of sets is denoted by |A|, # A, card (A), or n (A). A set is said to be finite if -

• It is an empty set, or,
• if there is one to one correspondence between the elements in the set

Note: A 1 - 1 correspondence between two sets A and B is another name for a function f: A -> B that is 1-1 and onto.

Now, let's understand it via examples.

Exmaple1: Suppose there is a set A that contains all English alphabets. Then the cardinality of set A is 26 because there are 26 letters in the English alphabet.

So, n(A) = 26.

Similarly, if there is a set of months in a year, it will have a cardinality of 12, as there are twelve months in a year.

Example2: Let P = {k, l, m, n}. Find its cardinality.

Solution: It is clear that the given set P is a finite set. So, the cardinality of the set P is equal to the number of elements in it. Therefore, the cardinality of the given set P is 4, as it contains four elements.

Hence, n(P) = 4.

So, it is easy to find the cardinality of finite sets. Now, let's discuss the cardinality of infinite sets.

Cardinality of Infinite sets

Before understanding the cardinality of infinite sets, first, understand infinite sets. A set that is not finite is called Infinite Sets. A set is infinite if it is not empty and also cannot be put into 1 - 1 correspondence with {1, 2, ……, n} for any n ∈ N. The number of elements in an infinite set is not countable and also cannot be represented in the roaster form. Therefore, the infinite set elements are represented by three dots (or ellipse).

Examples: The examples of infinity sets include the set of whole numbers, the set of all integers, and many more.

The cardinality of infinite sets is n(A) = ∞, where A is any infinite set. Its cardinality is infinite as it has unlimited number of elements.

Infinite sets are Countable infinite and Uncountable infinite.

Now, let's first discuss the Countable infinite sets and uncountable infinite sets.

Countable Infinite sets

A set is said to be countable infinite, if -

• it is finite, i.e., |A|< ∞
• If there is a one-to-one correspondence between the elements in the set and elements in the set of natural numbers N.

Any subset of a countable set is countable.

A countably infinite set is also known as Denumerable. A set that is either finite or denumerable is known as countable. As an instance, the set of a non-negative even integer is countable Infinite.

Furthermore, we designate the cardinality of countably infinite sets by ℵ₀ (pronounced as "aleph-naught" or "aleph null", where aleph is a letter in the Hebrew alphabet).

Suppose a set A has the same cardinality as the set of natural numbers, i.e., |A| = |N|. So, |A| = |N| = ℵ_0­. Thus, the cardinality of any countably infinite set is ℵ₀. The following sets have cardinality ℵ₀.

• Set of Even numbers
• Set of perfect squares, cubes, or perfect nth numbers
• Set of prime numbers, etc.

Uncountable infinite sets

A set that is not countable is called Uncountable Infinite Set or non-denumerable set, or simply Uncountable.

Example: Set R of all +ve real numbers less than 1 that can be represented by the decimal form 0.a1,a2,a3..... Where a1 is an integer such that 0 ≤ ai ≤ 9.

There are many characterizations of an uncountable infinite set. Some of them are listed as follows:

• If the set is neither finite nor equal to the cardinality of natural numbers, i.e., ℵ₀ (aleph-null).
• The given set has cardinality strictly greater than ℵ₀.

The example of an uncountable set is the set of all real numbers. And the more abstract example of an uncountable set is the set of all countable ordinal numbers.

Now, let's see some of the questions to find the cardinality of a set.

Question1- Suppose there are two sets A and B in which A = {1, 2, 3, 4, 5, 6} and B = {2, 4, 6, 8}. What is the cardinality of set A, the cardinality of A ⋃ B, the cardinality of A ⋂ B?

Solution- A = {1, 2, 3, 4, 5, 6}, B = {2, 4, 6, 8}

It is clear that the cardinality of set A is 6 as there are six elements in set A.

A ⋃ B = {1, 2, 3, 4, 5, 6} ⋃ {2, 4, 6, 8} = {1, 2, 3, 4, 5, 6, 8}

A ⋂ B = {1, 2, 3, 4, 5, 6} ⋂ {2, 4, 6, 8} = {2, 4, 6}

So the cardinality of A ⋃ B is 7 and the cardinality of A ⋂ B is 3.

Question2- What is the cardinality of the set M = {0} and N = {}?

Solution- The set M contains a single element, so the cardinality of set M is 1. On the other hand, the set N is empty, and as discussed above, the cardinality of an empty set is 0.