Exercise 1.2
PAGE No- 1.2
TYPE -1
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) $\{2,1,7\}$.
TYPE -2
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 $x \in B$.
PAGE No- 1.22
(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 $A \in C$
TYPE -3
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 $A \supseteq B, B \supseteq C \Rightarrow A \supseteq C$
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