Cardinal Numbers are Numbers that are used for counting something. They are also called the Cardinals. Cardinals are meant by how many of anything is existing in a group. In other words, cardinal numbers are a collection of ordinal numbers. Learn about the Cardinal Number of a Set Definition, Solved Examples explained in detail in the further modules.

## Cardinal Number of a Set – Definition

The Number of Distinct Elements present in a finite set is called the Cardinal Number of a Set. Usually, we define the size of a set using cardinality. The Cardinal Number of a Set A is denoted as n(A) where A is any set and n(A) represents the number of members in Set A.

Consider a set of even numbers less than 15.

Set A = {2, 4, 6, 8, 10, 12, 14}

As Set A has 7 elements, the Cardinal Number of the Set is n(A) = 7

Note:

(i) Cardinal number of an infinite set is not defined.

(ii) Cardinal number of the empty set is 0 since it has no element.

Also, Check:

### How to find the Cardinal Number of a Set?

1. Find the cardinal number of the following set

E = { x : x < 0, x ∈ N }

Solution:

x<0 means negative integers and they don’t fall under Natural Numbers.

Therefore, the above set will not have any elements

Cardinal Number of Set E is n(E) =0

2. Find the Cardinal Number of the Following Set

Q = { x : – 4 ≤ x ≤ 3, x ∈ Z }

Solution:

Given Q = { x : – 4 ≤ x ≤ 3, x ∈ Z }

x={-4, -3, -2, -1, 0, 1, 2, 3 }

Number of Elements in the above set is 8

Therefore, Cardinal Number of Set Q is n(Q) = 8

3. Find the Cardinal Number of the Set

A = { x : x is even prime number }

Solution:

Among all the prime numbers 2 is the only even prime number and the set has only one element

A ={2}

Cardinal Number of a Set n(A) = 1

4. Set D = {3, 4, 4, 5, 6, 7, 8, 8, 9}

Solution:

We know the cardinal number of a set is nothing but the number of distinct elements in the set

Cardinal Number of Set D is n(D) = 7

5. Find the Cardinal Number of a Set X = {letters in the word APPLE}

Solution:

Set X = {letters in the word APPLE}

We know the cardinal number of a set is nothing but the number of distinct elements in the set

x = {A, P, L, E}

Cardinal Number of Set n(X) = 4

6. Find the Cardinal Number of a Set

P = {x | x ∈ N and x^{2} <25}

Solution:

Given P = {x | x ∈ N and x^{2} <25}

Then P = {1, 2, 3, 4}

Cardinal Number of Set P is 4 and is denoted by n(P) = 4