KC Sinha Mathematics Solution Class 11 Chapter 1 Sets Exercise 1.3

 Exercise 1.3 

Page no- 1.44 

Type 1 

Question 1

If $A=\{2,3,4,5,6\}, B=\{3,4,5,6,7\}, C=\{4,5,6,7,8\}$, then find

(i) $(A \cup B) \cap(A \cup C)$ and $(A \cap B) \cup(A \cap C)$.

(ii) $A-(B \cup C)$

(iii) $A-(B-C)$

Question 2

If $S=\{0,1,2,3, \ldots, 9\}, A=\{0,1,2,3,4\}, B=\{1,2,3\}$, $C=\{5,6,7\}, D=\{5,7,8,9\}$, then find :

(i) $B-A$

(ii) $A^{\prime}$

(iii) $(C \cup D)$

(iv) $(D-C)^{\prime}$

(v) $C \cup S$

(vi) $A-(B-C)$.

Question 3

If $S=\{2,3,5,7,9\}, A=\{3,7\}, B=\{2,5,7,9\}$, then find

(i) $A \cap B^{\prime}$

(ii) $B \cap A^{\prime}$. Also verify that $\left(A \cup B Y=A^{\prime} \cap B^{\prime}\right.$.

Question 4

Let $A=\left\{x: x \in Z^{+}\right\}$, where $Z^{+}$denotes the set of all positive integers.

$B=\{x: x=3 n, n \in Z\}$

$C=\{x: x$ is a negative integer $\}$

$D=\{x: x$ is an odd integer $\}$

Find (i) $A \cap C$ (ii) $A-B$ (iii) $A \cup D$

Question 5

Let $A=\{3,6,12,15,18,21\}, B=\{4,8,12,16,20\}$, $C=\{2,4,6,8,10,12,14,16\}$ and $D=\{5,10,15,20\}$. Find

(i) $A \cap B$

(ii) $A \cup C$

(iii) $A-D$

(iv) $C \cap A$

(v) $D-A$

(vi) $B \cup C$

Page no- 1.45

Question 6

If $S=\{1,2,3, \ldots, n\}, A=\{x: x=2 n, n \in S\}$

$B=\{x: x=2 n-1 ; n \in S\}$ and $C=\{1,2,3, \ldots, n\}$, then find

(i) $(A \cap B Y$

(ii) $A^{\prime} \cap C$

(iii) $A \cap B^{\prime}$

(iv) $\boldsymbol{A}^{\prime}$

Question 7

If $A=\{3,5,7,9,11\}, \quad B=\{7,9,11,13\}$; $C=\{11,13,15\}$ and $D=\{15,177$; find

(i) $A \cap C \cap D$

(ii) $A \cap(B \cap C)$

(iii) $A \cap(B \cup D)$

(iv) $(A \cap B) \cap(B \cup C)$

(v) $(A \cup D) \cap(B \cup C)$

Question 8

If $A=\{5,6,7,8,9\}, B=\{2,4,6,8,10,12\}$ and $C=\{3,6,9,12\}$, then verify that $A-(B \cup C)=(A-B) \cap(A-C)$.

Question 9

Let $A=\{x: x$ is a natural number $\}, B=\{x: x$ is an even natural number $\}$,

$C=\{x: x$ is an odd natural number $\}$.

$D=\{x: x$ is a prime number $\}$. Find

(i) $A \cap B$

(ii) $A \cap C$

(iii) $B \cap D$

(iv) $C \cap D$

Question 10

If $A=\{x: x$ is a positive integer $<8$ and $x$ is a multiple of 3 or 5$\}$, $B=\left\{x: x^{3}-6 x^{2}+11 x-6=0\right\}$

$C=\{x: x$ is even number $\leq 7\}$, then show that

(i) $A \cap(B \cup C)=(A \cap B) \cup(A \cap C)$.

(ii) $A \cup(B \cap C)=(A \cup B) \cap(A \cup C)$.

(iii) $A \cap(B \cap C)=(A \cap B) \cap C$.

Question 11

On the real line, if $A=[0,3]$ and $B=[1,4]$ then find $A^{\prime}, B^{\prime}, A \cup B, A \cap B$ and $A-B$.

Question 12

Taking the set of all natural number $N$ as the universal set, write down the complements of the following sets :

(i) $\{x: x \in N$ and $x \geq 7\}$

(ii) $\{x: x \in N$ and $x$ is odd $\}$

(iii) $\{x: x \in N$ and $2 x+5=9\}$

(iv) $\{x: x \in N$ and $x$ is even $\}$

(v) $\{x: x$ is a prime number\}

(vi) $\{x: x \in N$ and $x$ is a perfect cube $\}$.

Question 13

Which of the following pairs of sets are disjoint ?

(i) $\{a, e, i, o, u\}$ and $\{c, d, e, f\}$

(ii) $\{1,2,3,4\}$ and $\{x: x$ is a natural number and $4 \leq x \leq 6\}$

(iii) $\{x: x$ is an even integer $\}$ and $\{x: x$ is an odd integer $\}$

Question 14

Give an example of sets $A, B, C$ such that $A \cap B \neq \phi$,

 $B \cap C \neq \phi$ and $A \cap C \neq \phi$, but $A \cap B \cap C=\phi$.

Type 2

Question 15

For sets $A, B$ and $C$, prove the following using the properties of set operations :

(i) $(A \cup B)-A=B-A$

(ii) $A-(B \cup C)=(A-B) \cap(A-C)$

(iii) $A \cup(B-A)=A \cup B$

(iv) $A-(B-C)=(A-B) \cup(A \cap C)$

(v) $(A-B)-C=A-(B \cup C)$

(vi) $A-(A-B)=B \Leftrightarrow B \subseteq A$

(vii) $A \cup(A \cap B)=A$

(viii) $(A \cap B) \cup(A-B)=A$

(ix) $A \cap(A \cup B)=A$

(x) $A \cup(B-A)=A \cup B$

(xi) $(A \cup B) \cap\left(A \cup B^{\prime}\right)=A$

Page no 1.46

Type 3

Question 16

Prove the following :

(i) $A \cap B \subseteq A$ and $A \cap B \subseteq B$

(ii) $A=B \Rightarrow A \cap C=B \cap C$

(iii) $C \subseteq A$ and $C \subseteq B \Rightarrow C \subseteq A \cap B$

(iv) $A \subseteq B \Rightarrow B^{\prime} \subseteq A^{\prime}$

(v) $(A \cup \phi) \cap(A \cap \phi)=\phi$

(vi) $A \cap(A \cup B)=A \cup(A \cap B)=A$

(vii) $A \subseteq B \Rightarrow C-B \subseteq C-A$

(viii) $A \subseteq \phi \Rightarrow A=\phi$

(ix) $A^{\prime} \cup B=S \Rightarrow A \subseteq B$

Question 17

Prove the following :

(i) $\left(A^{\prime}\right)=A$

(ii) $A-B=A-(A \cap B)=(A \cup B)-B$

(iii) $(A \cap B) \cup(A-B)=A$

(iv) $A \cap(A \cup B)=\phi$

(v) $A \cap B=A$ and $A \cup B=A \Rightarrow A=B$

(vi) $A-B=A \Leftrightarrow A \cap B=\phi$

(vii) $A \subseteq B \subseteq C \Rightarrow C-(B-A)=A \cup(C-B)$

(viii) $(A \cup B) \cap B^{\prime}=A \Leftrightarrow A \cap B=\phi$

Question 18

 Show that the following conditions are equivalent :

(i) $A \subseteq B$

(ii) $A \cap B=A$

(iii) $A-B=\phi$

(iv) $A \cup B=B$

Type 4

Question 19

Shade the following sets :

(i) $A^{\prime} \cap B^{\prime}$

(ii) $A^{\prime} \cup B^{\prime}$

(iii) $(A \cup B) \cap(A \cup C)$

(iv) $(A \cap B) \cup(A \cap C)$