Cardinal Numbers:  Cardinal numbers are used for counting different things. These are also known as cardinals. These are the natural numbers that start from \(1\) and go on sequentially and are not fractions. The word cardinals refer to ‘how many of anything is existing in a group. Like if we want to count the number of apples present in the basket, we must make use of these numbers, such as \(1, 2, 3, 4, 5,\)…. and so on. Numbers help to count the number of things; people present in the place or a group.

Set

Definition: A set is a collection of well-defined objects.

Here, the word ‘well defined’ means it should be possible to determine whether an object does or does not belong to a specific collection. 

The objects of the set are known as its elements or members. The symbol \(\in \) is used to mean “is an element of”, and the symbol \( \notin \) is used to denote “is not an element of”.

Example: In set \(A = \left\{={a,~b,~c} \right\};\) \(a, b\) and \(c\) are the elements of set \(A\).

We write, \(a \in A\), for \(“a\) is an element of \(A”\).

\(b \in A\), for \(“b\) is an element of \(A”\).

\(c \in A\), for \(“c\) is an element of \(A”\).

And \(d \in A\), for \(“d\) is not an element of \(A”\).

Ways to Represent a Set

To represent a set, we use the following methods:

Description Method

In this method, we write the (well-defined) description of the elements of the set and this description is enclosed in curly brackets.

Example: 

1. The ‘set of prime numbers less than \(10’\) is written as {prime numbers less than \(10\)}.
Note that \(0 \in \) {whole numbers less than \(10\)} while \(10 \notin \) {whole numbers less than \(10\)}.
2. The set of even integers is written as {even integers}.
Note that \(8 \in \) {even integers} while \(5 \notin \) {even integers}.
3. The set of colours of the rainbow is written as {colours of rainbow}.
Note that blue \( \in \) {colour of rainbow} while black \( \notin \) {colours of rainbow}.

Roster Method

In this method, we list all the elements of the set and separate them by commas. The pair of flowers (curly) brackets enclose the list. 

Example: The set \(A\) of even whole numbers less than \(10\) in the roster method is written as:

\(A = \left\{ {0,~2,~4,~6,~8} \right\}\).

Note that \(0,\,6 \in A\) while \(10 \notin A\).

\(A = \left\{ {a,~e,~i,~o,~u} \right\}\) represents the set of vowels.

Set Builder Form

In set-builder form, we write a variable (say \(x\)) representing any number of the set followed by a property satisfied by each element of the group and enclose it in flower (curly) brackets.

If \(A\) is the set consisting of elements \(x\) having property \(p\), we write:

\(A = \{ x|x\) has property \(p\} \)

Which is read as ‘the set of elements \(x\) such that \(x\) has property \(p’\).

The symbol \(‘|\)’ stands for the words ‘such that’. Sometimes, we use the logo \(‘:’\) in place of the symbol \(‘|’\).

Example: The set \(A\) of whole numbers less than \(10\) in the set-builder form is written as 

\(A = \{ x|x \in W,~x < 10\} \)

It is read as ‘the set of elements \(x\) such that \(x\) belong to \(W\) and \(x\) is less than \(10’\).

Venn Diagram

The relationship between the sets can be represented by means of diagrams which are known as Venn Diagrams.

The Venn Diagram consists of rectangles and closed curves, usually circles. The universal set is represented normally by a rectangle and the subsets of it by circles.

In Venn Diagrams, the elements of a set are commonly written in their respective circles.

Example: \(U = \left\{ {1,~2,~3,~ \ldots .,~10} \right\}\) is the universal set of which \(A = \left\{ {~2,~4,~6,~8,~10} \right\}\) is a subset.

Cardinal Number

Definition: The symbol to represent the number of elements in a set \(A\) is \(n (A)\). it is known as a cardinal number of set \(A\).

A set may have

1. No elements
2. One or more elements with
a. A finite number of elements.
b. An infinite number of elements.

Cardinal Numbers for Different Sets

According to the number of elements contained in a set, we define different types of sets as follows:

Finite Set

A set that contains a limited (counted) number of different elements is called a finite set. In other words, a set is called a finite set if the counting of its various parts comes to an end.

Example: \(A = \left\{ {~a,~b,~c,~d,~e} \right\}\) is a finite set having five elements. The cardinal number of set \(A\), denoted by \(n(A)\) is \(5\).

\(B = \{ x|x \in W,~x < 7\} \) is a finite set having seven elements. The cardinal number of set \(B\), denoted by \(n(B)\) is \(7\).

\(C = \left\{ {1,~3,~5,~ \ldots .,~10} \right\}\) is a finite set. The cardinal number of set \(C\), denoted by \(n(C)\) is \(101\).

Infinite Set

A set that contains an unlimited (uncountable) number of different elements is called an infinite set. In other words, a set is called if the counting of its various elements does not come to an end.

Example: the set of natural numbers \(N = \left\{ {~1,~2,~3,~4,~…..} \right\}\)

The set whole numbers \(W = \left\{ {~0,~1,~2,~3,~…..} \right\}\)

The set of integers \(I = \left\{ {~-3,~-2,~-1,~0,~1,~2,~3,~……} \right\}\)

Each of the above sets is an infinite set. Hence, the cardinal number for each of them is an infinite number.

Empty Set

The set containing no elements is a blank or void, or null set. It is denoted by the symbol \(\phi \) or \(\left\{ ~ \right\}\).

Example: The set of cats with two tails is the empty set.

The set of students in your class aged \(40\) years is the empty set.

The set of prime numbers, which are irrational numbers, is the empty set.

Each of the above sets is an empty set. Hence, the cardinal number for each of them is \(0\) (zero).

Singleton Set

A set containing one element is called a singleton (or unit) set.

Example: \(\left\{ { – 3} \right\}\)

\(\{ x:x \in W,~x < 1\} \)

\(\{ x:x\) is capital of India \(\} \)

Each one of these is a singleton set. Hence, the cardinal number for each of them is \(1\) (one).

Equal Set

Two sets are called equal sets if they have the same elements.

If sets \(A\) and \(B\) are equal, we write \(A=B\); and if sets \(A\) and \(B\) are not equal, we write \(A≠B\).

Example: If \(A = \left\{ {~a,~b,~c} \right\}\) and \(B = \left\{ {~b,~a,~c} \right\}\), then \(A=B\) because the elements in a set can be repeated or rearranged. Here, both sets \(A\) and \(B\) have \(3\) elements each. Hence, their cardinal numbers are the same. \(n\left(A\right) = n\left(B\right)\).

If \(A = \left\{ {~-3,~-2,~-1,~0,~1,~2,~3,~……} \right\}\) and \(B = \left\{ {x\mid x \in I,\, {x^2} < 10} \right\}\), then \(A=B\) because if we write \(B\) in tabular form, we get the same elements. Here, both sets \(A\) and \(B\) have \(7\) elements each. Hence, their cardinal numbers are the same. \(n\left(A\right) = n\left(B\right)\).

Equivalent Set

Two (finite) sets are called equivalent sets if they have the same number of elements. Thus, two finite sets, \(A\) and \(B\) are comparable sets, written as \(A↔B\) (read as \(‘A\) is equivalent to \(B’\)), if \(n\left(A\right) = n\left(B\right)\).

Example: If \(A = \left\{ {~a,~e,~i,~o,~u } \right\}\) and \(B = \left\{ {~1,~2,~3,~4,~5} \right\}\), then \(n(A)=5=n(B)\).

Hence, the cardinal number of equivalent sets are equal.

Cardinal Number Formulas

• For two disjoint sets, \(n(A \cup B) = n(A) + n(B)\)
• For two overlapping sets, \(n(A \cup B) = n(A) + n(B) – n(A \cap B)\)
• For three disjoint sets, \(n(A \cup B \cup C) = n(A) + n(B) + n(C)\)
• For three overlapping sets, \(n(A \cup B \cup C) = n(A) + n(B) + n(C) – n(A \cap B) – n(B \cap C) – n(C \cap A) + n(A \cap B \cap C)\)

Solved Examples on Cardinal Numbers

Q.1. Write the given set in set-builder form: \(\left\{ {0,~1,~4,~9,~16,~25,~36} \right\}\).
Ans: The given numbers are a perfect square of the integers from \(-6\) to \(+6\).
\(\therefore \) Given set \( = \{ x:x = n2,~\,n \in I\) and \( – 6 \leqslant n \leqslant  + 6\} \).

Q.2. Write the given set in set-builder form: \(\left\{ {-10,~-5,~0,~5,~10,~15,~20,~25} \right\}\).
Ans: The given numbers are multiples of \(5\) lying between \(-10\) and \(25\) (both inclusive).
\(\therefore \) Given set \( = \{ x|x = 5n,~\,n \in I\) and \( – 2 \leqslant n \leqslant 5\} \).

Q.3. Write the given set in roster form and find the cardinal number.
\(A\)={vowels in the word TELEVISION}
Ans: Clearly, TELEVISION has \(2 Es, 2 Is\) and one \(O\), but while listing out elements in the roster form, we do not repeat elements.
In roster form, \(A = \left\{ {E,~\,I,\,~O} \right\}\) and \(n\left( A \right) = 3\)

Q.4. Write the given set in roster form and find the cardinal number: \(B=\){factors of \(24\)}
Ans: We know that,
The factors of \(24\) are \(1, 2, 3, 4, 6, 8, 12, 24\)
In roster form, \(B = \left\{ {~1,~2,~3,~4,~6,~8,~12,~24} \right\}\). Since it has eight elements, \(n\left( B \right) = 8\).

Q.5. Write the given set in roster form and find the cardinal number: 
\(C = \{ x|x = 4n + 3,~\,0 < n < 7,~\,n \in N\} \)
Ans: To get the elements of this set, we shall have to substitute \(n=1, 2, 3, 4, 5\) and \(6\) in the given equation and list out all the values of \(x\) that we get.
For \(n=1, x=4×1+3=7\);
For \(n=2, x=4×2+3=11\), and so on.
Thus, we have \(C = \left\{ {~7,~11,~15,~19,~23,~27} \right\}\) and \(n\left( C\right) = 6\).

Summary

In this article, we discussed the definition of a set, different ways a set can be represented, the definition of cardinal number and the cardinal number for different types of sets. After reading this article, the students will have a fair understanding of the cardinal number related to the set theory.

Frequently Asked Questions (FAQs) on Cardinal Numbers

Q.1. Is \(21\) a cardinal number?
Ans: Cardinal numbers are the numbers that are used for counting different things. These are also known as cardinals. These are whole numbers that start from \(0\) and goes on sequentially and are not fractions. So, yes, \(21\) is the cardinal number.

Q.2. What is the ordinal number of \(100\)?
Ans: An ordinal number defines the position of an item or a number in a series, such as ‘first’, ‘seventh’, ‘eleventh’ etc. Hence, the ordinal number of \(100\) is written as “one hundredth” or “the hundredth”.

Q.3. What is the Cardinal number of a null set?
Ans: A null set does not contain any element in it. Hence, the cardinal number of a null set is \(0\) (zero).

Q.4. Is \(11\) a cardinal number?
Ans: Cardinal numbers are the numbers that are used for counting different things. These are also known as cardinals. These are whole numbers that start from 0 and goes on sequentially and are not fractions. So, yes, \(11\) is the cardinal number.

Q.5. What does Cardinal number mean in math?
Ans: In mathematics, cardinal numbers are a generalization of the whole numbers used to measure the cardinality (size) of sets.

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