Cardinal Properties of Sets: Definition, Properties, Formulas, Examples

Sets are a collection of well-defined elements that do not vary from person to person. The number of elements or objects present in a finite set is known as its cardinal number.

All the elements in a set are written between the curly braces, and a comma is used to separate the elements. Cardinality means the number of elements present in the given set. In this article, we will learn more about the cardinal properties of sets, formulas and examples.

What are Cardinal Numbers?

Cardinal numbers are also counting numbers. By counting numbers, we mean natural numbers. We can call cardinal numbers as just cardinals. The cardinality of sets is the size of the set. Cardinality is the number of elements present in a finite set that describes the size of the set. The smallest cardinal number is \(1\), as we start with \(1\) for counting anything, so \(0\) is not a cardinal number.

Example:

Consider two sets, set \(A = \left\{{1,3,~4,~6,~7} \right\}\) and set \(B = \left\{{2,3,~5,~6,~8,~9} \right\}\)

The number of elements in set \(A=5\), therefore the cardinality of set \(A\) is \(5\).

Similarly, the cardinality of set \(B\) is \(6\).

Cardinal number gives us an idea regarding the number of elements presents in a set, such as one, two, three, four, five, etc.

There are other two numbers, namely ordinal and nominal, apart from cardinal numbers. While the cardinal numbers tell us the size of the set, ordinal and nominal numbers describes the position of things and the name, respectively.

In the above picture, there are \(7\) cars. On the car door, the car numbers have been written. Car number “\(99\)” (with the yellow roof) is in the first position:

• \(7\) is a cardinal number (it tells the number)
• \({1^{{\text{st}}}}\) is an ordinal number (it tells position)
• “\(99\)” is a nominal number (it is just a name for the car)

Definition of Cardinal Number

In a set, the cardinal number is the total number of elements present in it. In other words, the number of distinct elements present in a set is the set’s cardinal number. The cardinal number is represented as \(n(A)\) for set \(A\).

Cardinal Properties of Sets: Definition

The basic properties of sets are union and intersection of two or three sets. We learnt that the cardinals are the size of a set of well-defined elements available in it. Cardinal properties of sets deal with the sets union, intersection along with the number of sets properties. Some of the formulas that describe the properties of sets are given in the next section.

Cardinal Number of a Set Formula

i) Union of Disjoint Sets:

If \(A\) and \(B\) are two finite sets and if \(A \cap B=\emptyset\), then \(n(A \cup B)=n(A)+n(B)\).

The union of the disjoint sets \(A\) and \(B\) represented by the Venn diagram is given by \(A \cup B\), and it can be seen that \(A \cap B=\emptyset\) because there are no element common in the sets.

ii) Union of Two Sets

If \(A\) and \(B\) are two sets with finite elements, then \(n(A \cup B)=n(A)+n(B)-n(A \cap B)\)

In the above figure, the shaded regions indicate the different disjoint sets, i.e. \(A-B, B-A\) and \(A \cap B\) are three disjoint sets as shown, and the sum is represented by \(A \cup B\). Hence, \(n(A \cup B)=n(A-B)+n(B-A)+n(A \cap B)\).

iii) Union of Three Sets

If \(A, B\) and \(C\) are three finite sets, then,

\(n\left( {A \cup B \cup C} \right) = n\left( A \right) + n\left( B \right) + n\left( C \right) – n\left( {A \cap B} \right) – n\left( {B \cap C} \right)\)
\( – n\left( {A \cap C} \right) + n\left( {A \cap B \cap C} \right)\)

It can be observed from the Venn diagram that the union of the three sets will be the sum of the cardinal number of set \(A\), set \(B\), set \(C\) and the repeated elements of the three sets, excluding the repeated elements of sets taken in pairs of two.

Example:

Let us solve an example to understand the concept discussed above.

There are \(180\) students in class XII. \(100\) of the study science, \(50\) students study commerce, and \(30\) students study both science and commerce. Find the number of students who study science but not commerce

Solution: The total number of students represents the cardinal number of the set. Let \(A\) denote the set of students studying science, and set \(B\) represent the students studying commerce.

Therefore,

\(n(U)=180\)

\(n(A)=100\)

\(n(B)=50\)

\(n(A \cap B)=30\)

Here, we need to find the difference of sets \(A\) and \(B\).

\(n(A)=n(A-B)+n(A \cap B)\)

\(n(A-B)=n(A)-n(A \cap B) \Rightarrow n(A-B)=100-30=70\)

The number of students who study science but not commerce is \(70\).

Solved Examples on Cardinal Properties of Sets

Q.1. If \(C\) and \(D\) are two sets, \(C \cup D\) has \(36\) elements, \(C\) has \(18\) elements, and \(D\) has \(22\) elements, how many elements does \(C \cap D\) have?
Ans: Given, \(n(C \cup D)=36, n(C)=18, n(D)=22\)
We know that \(n(C \cup D)=n(C)+n(D)-n(C \cap D)\)
So, \(36=18+22-n(C \cap D) 36=40-n(C \cap D)\) Therefore, \(n(C \cap D)=40-36=4\)
Therefore, \(C \cap D\) have \(4\) elements.

Q.2. If \(A\) and \(B\) are two finite sets such that \(n(A)=25, n(B)=11\) and \(n(A \cap B)=4\), find \(n(A \cup B)\).
Ans: Given that, \(n(A)=25, n(B)=11\) and \(n(A \cap B)=4\)
We know that \(n(A \cup B)=n(A)+n(B)-n(A \cap B)\)
\(=25+11-4\)
\(=36-4\)
\(=32\)
Therefore, \(n(A \cup B)=32\).

Q.3. \(600\) people are residing in an apartment. Out of these, \(340\) people take “The Hindu Times”, and \(270\) people take “The Economic Times”. \(70\) persons who take both the papers. Find the number of people who do not take either of the two papers.
Ans: Total people \(=600\)
\(340\) people take The Hindu. Among these, \(70\) people take both papers
so \((340-70)=270\) people take only The Hindu.
\(270\) people take The Economic Times. Among these \(70\) persons take both the papers
so, \((270-70)=200\) peoples take only The Economic Times.
so, the number of persons who do not take either of the two papers \(=600-(270+200+70)\)
\(=600-540\)
\(=60\)
Thus, \(60\) people do not take papers.

Q.4. In a group of \(80\) people, \(30\) like vanilla ice cream, \(20\) like strawberry and 9 like both strawberry and vanilla.
Find:
(a) how many like vanilla only?
(b) how many like strawberry only?
(c) how many like at least one of them?
(d) how many like none of them?
Ans: Given that the total number of people in the group \(=80\)
Let \(V\) be the set of people who like vanilla ice cream, \(S\) be the set of people who like strawberry ice cream.
Number of people like vanilla, \(n(V)=30\)
Number of people who like strawberry, \(n(S)=20\)
Number of people who like vanilla and strawberry, \(n(V \cap S)=9\)
(a) The number of people who like vanilla only means \(n(V-S)\)
Using the relation, \(n(V-S)=n(V)-n(V \cap S)\)
\(=30-9\)
\(=21\)
So, \(21\) people like vanilla ice cream only.
(b) The number of people who like strawberry only means \(n(S-V)\)
\(n(S-V)=n(S)-n(V \cap S)\)
\(=20-9\)
\(=11\)
So, \(11\) people like only strawberry ice cream.
(c) The number of people who like at least vanilla or strawberry ice cream means \(n(V \cup S)\)
\(n(V \cup S)=n(V)+n(S)-n(V \cap S)\)
\(=20+30-9\)
\(=50-9\)
\(=41\)
Therefore, \(41\) people like at least one of them.
(d) The number of people who like none of them \(=\) total number of people in the group \(-\) number of people who like at least one of them
\(=80-41\)
\(=39\)
Therefore, \(39\) people like none of them.

Q.5. Help Arya, to calculate the number of vowels in “MATHEMATICS“.
Ans: We know that \(a, e, i, o\), and \(u\) are the vowels. In the word, “MATHEMATICS”, \(a, e\) and \(i\) are used. Therefore, \(3\) vowels are used to form the word.
\(V = \left\{ {a,\,e,\,i} \right\}\)
There are three elements. Thus, \(3\) is the cardinal number.

Summary

This article is written to discuss the cardinal properties of sets. We learnt the meaning and definition of cardinal numbers and cardinal sets. Then we studied the various properties of cardinal sets. Further, we learnt the formulas related to the cardinality of sets. Various examples have been solved to make the students learn the concept well.

FAQs on Cardinal Properties of Sets

Q.1. How do you find the cardinal number of a set?
Ans: The number of objects or elements in the given set is called the cardinal number of the set. The cardinal number of a set with no elements in it is always zero. If the elements of a set \(A\) are \(\left\{{5,7,13,18,23,32,39}\right\}\), then the cardinal number for \(A\) is \(7\), and it is represented as \(n(A)=7\).

Q.2. What are the various cardinal properties of sets?
Ans: The various cardinal properties of \(n(A \cup B)=n(A)+n(B)-n(A \cap B), n(A-B)=n(A)-n(A \cap B)\) and if \(A \cap B=\phi\), then \(n(A \cup B)=n(A)+n(B)\).

Q.3. What is the cardinality of a set?
Ans: The cardinality of a set \(A\) is represented as \(n(A)\). It counts the number of elements in a set. We can find the cardinality only for sets with a finite number of elements.

Q.4. What does the cardinal number mean in sets?
Ans: The number of distinct elements present in a finite set is called its cardinal number. It is denoted as \(n(A)\) and read as “the number of elements of the set \(A\)”.
Example: Set \(B = \left\{{{\text{Mumbai}},{\text{Bangalore}},{\text{Ahmedabad}},{\text{Hyderabad}}}\right\}\) has \(4\) elements. Therefore, the cardinal number of set \(B=4\).

Q.5. What is a cardinality example?
Ans: Suppose a set \(A\) has a finite number of elements. Its cardinality is simply the number of elements in that set \(A\). For example, if \(A = \left\{ {2,~4,~6,~8,~10,~12,~14} \right\}\), then \(n(A)=7\).

Q.6. What is the biggest cardinal number?
Ans: There are infinite natural numbers. Therefore the cardinal numbers are also infinite. As we do not know the biggest natural number, so does the biggest cardinal number.

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