Chapter 1 Sets EXERCISE 1.4
EXERCISE 1.4
1. Find the union of each of the following pairs of sets :
(i) $X=\{1,3,5\} \quad Y=\{1,2,3\}$
(ii) $A=[a, e, i, o, u] \quad B=\{a, b, c\}$
(iii) $A=\{x: x$ is a natural number and multiple of 3 $\}$
$B=\{x: x$ is a natural number less than 6 $\}$
(iv) $A=\{x: x$ is a natural number and $1<x \leq 6\}$
$B=\{x: x$ is a natural number and $6<x<10\}$
(v) $A=\{1,2,3\}, B=\phi$
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Answer :
(i) $X=\{1,3,5\} Y=\{1,2,3\}$
$X \cup Y=\{1,2,3,5\}$
(ii) $A=\{a, e, i, o, u\} B=\{a, b, c\}$
$A \cup B=\{a, b, c, e, i, o, u\}$
(iii) $ A=\{x: x\}$ is a natural number and multiple of 3 $=\{3,6,9 \ldots \}$
As $ B = \{x: x$ is a natural number less than 6 $ \}=\{1,2,3,4,5,6\}$
$A \cup B=\{1,2,4,5,3,6,9,12 \ldots\}$
$\therefore A \cup B=\{x: x=1,2,4,5\}$ or a multiple of 3
(iv) $A=\{x: x.$ is a natural number and $1<x - 6\}=\{2,3,4,5,6\}$
$B=\{x: x$ is a natural number and $6<x<10\}=\{7,8,9\}$
$A \cup B=\{2,3,4,5,6,7,8,9\}$
$\therefore A B B=\{x: x \in N$ and $1<x<10\}$
(v) $A=\{1,2,3\}, B=\Phi$
$A \cup B=\{1,2,3\}$
2. Let $A=\{a, b\}, B=\{a, b, c\}$. Is $A \subset B$ ? What is $A \cup B$ ?
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Answer:
Here, $A=\{a, b\}$ and $B=\{a, b, c\}$
Yes, A $\subset$ B.
$A \cup B=\{a, b, c\}=B$
3. If $A$ and $B$ are two sets such that $A \subset B$, then what is $A \cup B$ ?
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Answer :
If $A$ and $B$ are two sets such that $A \subset B$, then $A \cup B=B$.
4. If $A=\{1,2,3,4\}, B=\{3,4,5,6\}, C=\{5,6,7,8\}$ and $D=\{7,8,9,10\}$; find
(i) $A \cup B$
(ii) $A \cup C$
(iii) $B \cup C$
(iv) $B \cup D$
(v) $A \cup B \cup C$
(vi) $A \cup B \cup D$
(vii) $B \cup C \cup D$
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Answer :
$A=\{1,2,3,4\}, B=\{3,4,5,6\}, C=\{5,6,7,8\}$ and $D=\{7,8,9,10\}$
(i) $A \cup B=\{1,2,3,4,5,6\}$
(ii) $A \cup C=\{1,2,3,4,5,6,7,8\}$
(iii) $B \cup C=\{3,4,5,6,7,8\}$
(iv) $B \cup D=\{3,4,5,6,7,8,9,10\}$
(v) $A \cup B \cup C=\{1,2,3,4,5,6,7,8\}$
(vi) $A \cup \cup B \cup D=\{1,2,3,4,5,6,7,8,9,10\}$
(vii) $B \cup C \cup \cup=\{3,4,5,6,7,8,9,10\}$
5. Find the intersection of each pair of sets of question 1 above.
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Answer :
(i) $X=\{1,3,5\}, Y=\{1,2,3\}$
$X \cap Y=\{1,3\}$
(ii) $A=\{a, e, i, o, u\}, B=\{a, b, c\}$
$A \cap B=\{a\}$
(iii) $ A= x $ : x is a natural number and multiple of 3$ =(3,6,9 )$
$B=\{x : x \text{ is a natural number less than} 6 \} =\{1,2,3,4,5\}$
$\therefore A \cap B=\{3\}$
(iv) $A=\{x.$ : xis a natural number and $.1<x < 6 \}=\{2,3,4,5,6\}$
$B=\{x$ : xis a natural number and $6<x<10\}=\{7,8,9\}$
$A \cap B=\Phi$
(v) $A=\{1,2,3\}, B=\Phi$
$A \cap B=\Phi$
6. If $A=\{3,5,7,9,11\}, B=\{7,9,11,13\}, C=\{11,13,15\}$ and $D=\{15,17\}$; find
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(i) $A \cap B$
(ii) $B \cap C$
(iii) $A \cap C \cap D$
(iv) $A \cap C$
(v) $B \cap D$
(vi) $A \cap(B \cup C)$
(vii) $A \cap D$
(viii) $A \cap(B \cup D)$
(ix) $(A \cap B) \cap(B \cup C)$
(x) $(A \cup D) \cap(B \cup C)$
7. If $A=\{x: x$ is a natural number $\}, B=\{x: x$ is an even natural number $\}$ $C=\{x: x$ is an odd natural number $\}$ and $D=\{x: x$ is a prime number $\}$, find
(i) $A \cap B$
(ii) $A \cap C$
(iii) $A \cap D$
(iv) $B \cap C$
(v) $B \cap D$
(vi) $C \cap D$
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Answer :
(i) $A \cap B=\{7,9,11\}$
(ii) $B \cap C=\{11,13\}$
(iii) $A \cap C \cap D=\{A \cap C\} \cap D=\{11\} \cap\{15,17\}=\Phi$
(iv) $A \cap C=\{11\}$
(v) $B \cap D=\Phi$
(vi) $A \cap(B \cup C)=(A \cap B) \cup(A \cap C)$
$=\{7,9,11\} \cup\{11\}=\{7,9,11\}$
(vii) $A \cap D=\Phi$
(viii) $A \cap(B \cup D)=(A \cap B) \cup(A \cap D)$
$=\{7,9,11\} \cup \Phi=\{7,9,11\}$
(ix) $(A \cap B) \cap(B \cup C)=\{7,9,11\} \cap\{7,9,11,13,15\}=\{7,9,11\}$
(x) $(A \cup D) \cap(B \cup C)=\{3,5,7,9,11,15,17\} \cap\{7,9,11,13,15\}$
$=\{7,9,11,15\}$
8. Which of the following pairs of sets are disjoint
(i) $\{1,2,3,4\}$ and $\{x: x$ is a natural number and $4 \leq x \leq 6\}$
(ii) $\{a, e, i, o, u\}$ and $\{c, d, e, f\}$
(iii) $\{x: x$ is an even integer $\}$ and $\{x: x$ is an odd integer $\}$
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Answer :
(i) $\{1,2,3,4\}$
$\{x: x.$ is a natural number and $4 \underset{-}{<} x \underset{-}{<} 6 \}=\{4,5,6\}$
Now, $\{1,2,3,4\} \cap\{4,5,6\}=\{4\}$
Therefore, this pair of sets is not disjoint.
(ii) $\{a, e, i, o, u\} \cap\{c, d, e, f\}=\{e\}$
Therefore, $\{a, e, i, o, u\}$ and $\{c, d, e, f\}$ are not disjoint.
(iii) $\{x: x$ is an even integer $\} \cap\{x: x$ is an odd integer $\}=$
$\Phi$ Therefore, this pair of sets is disjoint.
9. If $A=\{3,6,9,12,15,18,21\}, B=\{4,8,12,16,20\}$, $C=\{2,4,6,8,10,12,14,16\}, D=\{5,10,15,20\}$; find
(i) $A-B$
(ii) $A-C$
(iii) $A-D$
(iv) $B-A$
(v) $C-A$
(vi) $D-A$
(vii) $B-C$
(viii) $B-D$
(ix) $C-B$
(x) $D-B$
(xi) $C-D$
(xii) $D-C$
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Answer :
(i) $A-B=\{3,6,9,15,18,21\}$
(ii) $A-C=\{3,9,15,18,21\}$
(iii) $A-D=\{3,6,9,12,18,21\}$
(iv) $B-A=\{4,8,16,20\}$
(v) $C-A=\{2,4,8,10,14,16\}$
(vi) $D-A=\{5,10,20\}$
(vii) $B-C=\{20\}$
(viii) $B$ - D $=\{4,8,12,16\}$
(ix) $C-B=\{2,6,10,14\}$
(x) $D-B=\{5,10,15\}$
(xi) $C$ - D $=\{2,4,6,8,12,14,16\}$
(xii) $D-C=\{5,15,20\}$
10. If $X=\{a, b, c, d\}$ and $Y=\{f, b, d, g\}$, find
(i) $X-Y$
(ii) $Y-X$
(iii) $X \cap Y$
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Answer :
(i) $X-Y=\{a, c\}$
(ii) $Y-X=\{f, g\}$ (iii) $X \cap Y=$ $\{b, d\}$
11. If $\mathbf{R}$ is the set of real numbers and $\mathbf{Q}$ is the set of rational numbers, then what is $\mathbf{R}-\mathbf{Q}$ ?
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Answer :
$R$ : set of real numbers
Q: set of rational numbers
Therefore, $R-Q$ is a set of irrational numbers.
12. State whether each of the following statement is true or false. Justify your answer.
(i) $\{2,3,4,5\}$ and $\{3,6\}$ are disjoint sets.
(ii) $\{a, e, i, o, u\}$ and $\{a, b, c, d\}$ are disjoint sets.
(iii) $\{2,6,10,14\}$ and $\{3,7,11,15\}$ are disjoint sets.
(iv) $\{2,6,10\}$ and $\{3,7,11\}$ are disjoint sets.
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Answer :
(i) False
As $3 \in\{2,3,4,5\}, 3 \in\{3,6\}$
$\Rightarrow\{2,3,4,5\} \cap\{3,6\}=\{3\}$
(ii) False
As $a \in\{a, e, i, o, u\}, a \in\{a, b, c, d\}$
$\Rightarrow\{a, e, i, o, u\} \cap\{a, b, c, d\}=\{a\}$
(iii) True
As $\{2,6,10,14\} \cap\{3,7,11,15\}=\Phi$
(iv) True
As $\{2,6,10\} \cap\{3,7,11\}=\Phi$
