KC Sinha Mathematics Solution Class 11 Chapter 1 Sets Exercise 1.2

 Exercise 1.2 

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Question 1 

If $A=\{x: x$ is a letter in the word 'able' $\}$

$B=\{x: x$ is a letter in the word 'lable' $\}$

$C=\{x: x$ is letter in the word 'lab' $\}$

which of the following are true?

(i) $A=B$ (ii) $A \subset B$

(iii) $C \subseteq B$

Question 2

If $A=\left\{x: x \in Q\right.$ and $\left.x^{2}+3 x-10=0\right\}$

$B=\left\{x: x \in I\right.$ and $\left.-5 \leq x<\frac{17}{8}\right\}$, then show that $A \subseteq B$.

Question 3

Find the proper subsets of the following sets :

(i) $\phi$

(ii) $\{p, q, r\}$

(iii) $\{\{\phi\}\}$

(iv) $\{\{1\}, 1,2\}$

Question 4

Let $A=\{1,2,3,4\}, B=\{1,2,3\}$ and $C=\{2,4\}$

Find all sets $X$ satisfying the following pair of conditions

(i) $X \subseteq B, X \neq B$ and $X \mathscr{C}$

(ii) $X \subseteq B, X \subseteq C C$.

Question 5

Find the power sets of the following sets :

(i) $\{0\}$

(ii) $\{1,\{2,3\}\}$,

(iii) $\{2,1,7\}$.

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Question 6

Which of the following statements are true and which of them are false :

Given $A=\{4,11,17,21\}$.

(i) $4 \in\{4\}$

(ii) $\{11\} \in A$

(iii) $21 \notin A$

(iv) $11 \subset A$

(v) $\{11\} \subset A$

(vi) $\{\phi\} \subset A$

Question 7

If $A=\{1,2,\{3,4\},\{5\}\}$, then test which of the following are true :

(i) $\phi \in A$

(ii) $\{3,4\} \subset A$

(iii) $\{5\} \in A$

(iv) $\{2\} \in A$.

Question 8

Which of the following statements are correct :

(i) $\{\phi\}=\{0\}$

 (ii) $\{a, b, c\}=\{b, a, c\}$ 

(iii) $\{1,2,\{3\}\}=\{\{1\}, 2,3\}$

(iv) $\{\{1\}, 2,\{3\})=\{\{1\},\{3\}, 2\}$

 (v) $\{\phi\} \in\{\{\phi\}\}$

(vi) $\{\{1,2\},\{2\},\{2,3\}\}=\{\{1\},\{1,2\},\{3\}\}$

 (vii) $\{3\} \in\{2,3,4\}$

(viii) $\phi \subset\{a,\{a\}\}$ (ix) $a \in\{\{a\},\{\{a\}\}\}$ (x) $\{a\} \in\{a,\{a\}\}$

Question 9

Which of the following statements are true ?

(i) $S \subseteq P(S)$

(ii) $\{S\} \in P(S)$

(iii) $S \subset P(S)$

(iv) $\{S) \subset P(S)$

Question 10

Which of the following are true :

(i) Let $A$ and $B$ are two sets.

If $x \in A$ and $B \subseteq A$, then $x \in B$.

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(ii) Let A. $B$, $C$ be three sets.

If $A \subset B$ and $B \in C$, then $A \in C$.

Question 11

If $A=\{0,\{\phi\}, 1,\{1, \phi\}, 7\}$, then which of the following are true ?

(i) $7 \subset A$

(ii) $\{(\phi \mid\} \subset A$

(iii) $\{(7)\},(1)\} \subset \mathrm{A}$

(iv) $\{0, \phi\}\{1, \phi\}) \subset A$ (v) $(1\} \in A$

Question 12

Which of the following statements are true ?

(i) The set of all rectangles is contained in the set of all squares.

(ii) The sets $A=\{x: x$ is a lefter of the word 'LITTLE' $\}$

$B=\{x: x$ is a letter of the word 'TIT.E' $\mid$ are equal

(iii) The set of all cats is contained in the set of all animals.

Question 13

Examine which of the following statements are true or false :

(i) $\{a\} \in\{a, b, c\}$

(ii) Let $A=\{x: x$ is an even natural number which divides 6$\}$

$B=\{x: x$ is a natural number which divides 36$\}$

Then $A \subset B$

(iii) $\{a, b, c\} \subseteq\{a, b, c\}$

Question 14

Make correct statements by filling the blanks with suitable symbols $\in$ and $\mathrm{z}$

(i) $\{x: x$ is a square $\} \ldots . . \ldots\{x: x$ is a rectangle $\}$

(ii) $\{x: x$ is a circle in the plane $\} \ldots . .\{x: x$ is a circle in the plane with radius unity?

(iii) $\{2\} \ldots \ldots\{\{1,2\},\{3\}, 4\}$

(iv) $\{x: x$ is a student of class $\mathrm{XI}$ of your school $\} \ldots . .\{x: x$ is a student of your school

Question 15

In each of the following determine whether the statement is true or false . if is true, prove it and if it is false give an example 

(i) If $A \subseteq B$ and $B \subseteq C$, then $A \subseteq C$

(ii) If $A \nsubseteq B$ and $B \nsubseteq C$, then $A \nsubseteq C$

(iii) If $A \subseteq B$ and $B \in C$, then $A \in C$

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Question 16

Prove that $\{\{a\},\{a, b\}\}=\{\{c\},\{c, d\}\}$ iff $a=c$ and $b=d$

Question 17

(i) Prove that every subset of a finite set is finite

(ii) Prove that power set of a finite set is finite

Question 18

Prove that $A \supseteq B, B \supseteq C \Rightarrow A \supseteq C$