Set, Cardinality of Set, Cardinality of Two Sets, Cardinality of Three Sets, Example Questions with Answer (SEE)
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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.
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 Exercises
Reinforce your learning! Attempt these exercises to
build deep mastery and prepare for your quizzes.
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]
