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