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 Set SEE

Problems with Two Sets

1. 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.

  1. 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]
  2. Present the above information in a Venn diagram. [1U]
  3. Find the number of students who did not like any of the streams using a Venn diagram. [3A]
  4. If 24 students liked to study both the streams, is the condition of the Venn diagram changed? Give reason [1HA]

2. 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]

  1. 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]
  2. Present the above information in a Venn diagram. [1U]
  3. Find the number of people who liked I-phone only. [3A]
  4. Compare the number of people who liked both I-phone and Android phone and who do not like any of these two phones. [1HA]

 

Problems with Three Sets

3. In a survey of 120 people, it was found that 45 eat Grapes, 40 eat Pomegranates, 60 eat Amala, 10 eat Grapes and Pomegranates, 25 eat Grapes and Amala, 20 eat Pomegranates and Amala and 5 eat all three fruits.

  1. If A, B and C denote the set of people who eat Grapes, Pomegranates and Amala respectively, what does \(n(A \cap B \cap C)\) denote? [1K]
  2. Express the above information in a Venn diagram. [1U]
  3. Find the value of \(n(A \cup B \cup C)\). [3A]
  4. How many people in 24 people in an average do not eat all three fruits. Solve it. [1HA]

4. In a village of 140 houses, 70 believe in Hindu religion, 60 believe in Buddhism and 45 in other religions. Among them 17 houses don’t find any difference in Hindu and Buddhism, 18 houses don’t find any difference in Hindu and other religion and 16 houses don’t find any difference between Buddhism and other religions. If 6 houses don’t believe in any religion,

  1. A, B and C are the subsets of universal set U. If “A and B” and “B and C” are the overlapping sets and A and C are disjoint sets then what is the value of \(n(A \cap B \cap C)\)? [1K]
  2. Find how many houses do not find any difference in any religion. [3A]
  3. Identify the number of houses who don’t find any difference between Hindu and Buddhism. [1U]
  4. What is the percentage of houses who believe in other religion but not in Hindu and Buddhism? Calculate it. [1HA]

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