Force

Set

Unit: 1

Book Icon Class 10: Mathematics

Set, Cardinality of Set, Cardinality of Two Sets, Cardinality of Three Sets, Example Questions with Answer (SEE)

Set Introduction

A collection of well-defined objects is called a Set. The items in a set are called members of that set. 

 

Cardinality of Set

The number of members in a set is called the Cardinality of a set. If \(A\) is a set, then the cardinality of set \(A\) is represented by \(n(A)\). Note: you can remember \(n\) being \(\text{'number of members'}\) .

Example: Let’s say a set \(A = \{2,4,6,8,10\}\) then the cardinality of set \(A\) is \(n(A) = 5\). There are 5 members in the set \(A\). 

 

Cardinality of Two Sets

If \(A\) and \(B\) are overlapping sub-sets of the universal set \(U\) then, the following formulae can be derived.

\(\begin{aligned} & \text{1. } n(A \cup B) = n(A) + n(B) – n(A \cap B) \\ & \text{2. } n_0(A) = n(A) – n(A \cap B) \\ & \text{3. } n_0(B) = n(B) – n(A \cap B) \\ & \text{4. } n(A \cup B) = n_0(A) + n_0(B) + n(A \cap B) \\ & \text{5. } n(U) = n(A) + n(B) - n(A \cap B) + n(\overline{A \cup B}) \\ \end{aligned}\)

 

Cardinality of Three Sets

If \(A\) and \(B\) and \(C\) are overlapping subsets of the universal set \(U\) then, the following formulae can be derived.

\( \begin{aligned} & \text{1. } n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(C \cap A) + n(A \cap B \cap C) \\ & \text{2. } n(A \cup B \cup C) = n_0(A) + n_0(B) + n_0(C) + n_0(A \cap B) + n_0(B \cap C) + n_0(C \cap A) + n(A \cap B \cap C) \\ & \text{3. } n(U) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(C \cap A) + n(A \cap B \cap C) + n(\overline{A \cup B \cup C}) \\ & \text{4. } n(U) = n_0(A) + n_0(B) + n_0(C) + n_0(A \cap B) + n_0(B \cap C) + n_0(C \cap A) + n(A \cap B \cap C) + n(\overline{A \cup B \cup C}) \\ \end{aligned} \)

 

Practice Questions

Test your knowledge

Attempt these questions independently to reinforce your learning:

Q1.
The details obtained from a survey of 50 students of a school asking them about their further interests in studying the general stream or the technical stream are given below. [SEE 2080 KoP] 30 students liked to study the general stream. 24 students liked to study the technical stream. 9 students liked to study both streams.
  • a. Write the cardinality of the set of students who liked both of the streams by letting the sets of students who liked the general and technical stream by G and T respectively. [1K]
  • b. Present the above information in a Venn diagram. [1U]
  • c. Find the number of students who did not like any of the streams using a Venn diagram. [3A]
  • d. If 24 students liked to study both the streams, is the condition of the Venn diagram changed? Give reason [1HA]
Q2.
In a survey of 300 people, it was found that 150 people like I-phone and 200 people like Android phone. But 25 people did not like any of these two phones. [SEE MODEL 2080 A]
  • a. If I and A denote the sets of people who liked I-phone and Android phone respectively, write the cardinality of \(\overline{(I \cup A)}\). [1K]
  • b. Present the above information in a Venn diagram. [1U]
  • c. Find the number of people who liked I-phone only. [3A]
  • d. Compare the number of people who liked both I-phone and Android phone and who do not like any of these two phones. [1HA]

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