Class 11 Maths Chapter 1 – Sets Exercise 1.4

Exercise 1.4

Q1. Find the union of each of the following pairs of sets:

(The union of two sets consists of all the elements of both sets, the common elements being taken only once.)

(i) $X=\{1,3,5\}$ , $Y=\{1,2,3\}$

Sol.

X\cup Y=\{1,3,5\}\cup\{1,2,3\}\allowbreak=\{1,2,3,5\}

(ii) $A=\{a,e,i,o,u\}$ , $B=\{a,b,c\}$

Sol.

A\cup B=\{a,e,i,o,u\}\cup\{a,b,c\}\allowbreak=\{a,b,c,e,i,o,u\}

(iii) $A=\{x:x$ is a natural number and multiple of $\text{3}\}$ , $B=\{x:x$ is a natural number less than $\text{6}\}$

Sol. In roster form, $A=\{3,6,9,12,\ldots\}$ and $B=\{1,2,3,4,5\}$ .

A\cup B=\{3,6,9,12,\ldots\}\cup\{1,2,3,4,5\}\allowbreak=\{1,2,3,4,5,6,9,12,15,\ldots\}

∴ $A\cup B=\{x:x=1,2,4,5$ or $x$ is a multiple $\text{of }3,\ x\in N\}$

(iv) $A=\{x:x$ is a natural number $\text{and }1<x\le6\}$ , $B=\{x:x$ is a natural number $\text{and }6<x<10\}$

Sol. In roster form, $A=\{2,3,4,5,6\}$ and $B=\{7,8,9\}$ .

A\cup B=\{2,3,4,5,6\}\cup\{7,8,9\}\allowbreak=\{2,3,4,5,6,7,8,9\}

∴ $A\cup B=\{x:x$ is a natural number $\text{and }1<x<10\}$

(v) $A=\{1,2,3\}$ , $B=\phi$

Sol. $\phi$ has no element, so the union has only the elements of $A$ .

A\cup B=\{1,2,3\}\cup\phi=\{1,2,3\}=A

Q2. Let $A=\{a,b\}$ , $B=\{a,b,c\}$ . Is $A\subset B$ ? What is $A\cup B$ ?

Sol. Given, $A=\{a,b\}$ and $B=\{a,b,c\}$ .

Every element of $A$ belongs to $B$ .

∴ $A\subset B$

A\cup B=\{a,b\}\cup\{a,b,c\}=\{a,b,c\}=B

Q3. If $A$ and $B$ are two sets such that $A\subset B$ , then what is $A\cup B$ ?

Sol. Given, $A\subset B$ , so every element of $A$ is already in $B$ .

∴ $A\cup B=B$

Q4. 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) $B\cup C\cup D$

Sol. Given, $A=\{1,2,3,4\}$ , $B=\{3,4,5,6\}$ , $C=\{5,6,7,8\}$ and $D=\{7,8,9,10\}$ .

\textbf{(i)}\ A\cup B=\{1,2,3,4\}\cup\{3,4,5,6\}\allowbreak=\{1,2,3,4,5,6\}

\textbf{(ii)}\ A\cup C=\{1,2,3,4\}\cup\{5,6,7,8\}\allowbreak=\{1,2,3,4,5,6,7,8\}

\textbf{(iii)}\ B\cup C=\{3,4,5,6\}\cup\{5,6,7,8\}\allowbreak=\{3,4,5,6,7,8\}

\textbf{(iv)}\ B\cup D=\{3,4,5,6\}\cup\{7,8,9,10\}\allowbreak=\{3,4,5,6,7,8,9,10\}

(v) Using $A\cup B$ from (i),

A\cup B\cup C=(A\cup B)\cup C\allowbreak=\{1,2,3,4,5,6\}\cup\{5,6,7,8\}\allowbreak=\{1,2,3,4,5,6,7,8\}

(vi) Using $A\cup B$ from (i),

A\cup B\cup D=(A\cup B)\cup D\allowbreak=\{1,2,3,4,5,6\}\cup\{7,8,9,10\}\allowbreak=\{1,2,3,4,5,6,7,8,9,10\}

(vii) Using $B\cup C$ from (iii),

B\cup C\cup D=(B\cup C)\cup D\allowbreak=\{3,4,5,6,7,8\}\cup\{7,8,9,10\}\allowbreak=\{3,4,5,6,7,8,9,10\}

Q5. Find the intersection of each pair of sets:

(The intersection of two sets consists of the elements common to both sets.)

(i) $X=\{1,3,5\}$ , $Y=\{1,2,3\}$

Sol. $X\cap Y=\{1,3,5\}\cap\{1,2,3\}=\{1,3\}$

(ii) $A=\{a,e,i,o,u\}$ , $B=\{a,b,c\}$

Sol. $A\cap B=\{a,e,i,o,u\}\cap\{a,b,c\}=\{a\}$

(iii) $A=\{x:x$ is a natural number and multiple of $\text{3}\}$

Sol. In roster form, $A=\{3,6,9,12,\ldots\}$ and $B=\{1,2,3,4,5\}$ .

A\cap B=\{3,6,9,12,\ldots\}\cap\{1,2,3,4,5\}\allowbreak=\{3\}

(iv) $A=\{x:x$ is a natural number $\text{and }1<x\le6\}$

$B=\{x:x$ is a natural number $\text{and }6<x<10\}$

Sol. In roster form, $A=\{2,3,4,5,6\}$ and $B=\{7,8,9\}$ .

A\cap B=\{2,3,4,5,6\}\cap\{7,8,9\}\allowbreak=\phi

(v) $A=\{1,2,3\}$ , $B=\phi$

Sol. $A\cap B=\{1,2,3\}\cap\phi=\phi$

Q6. If $A=\{3,5,7,9,11\}$ , $B=\{7,9,11,13\}$ , $C=\{11,13,15\}$ and $D=\{15,17\}$ ; find

Given, $A=\{3,5,7,9,11\}$ , $B=\{7,9,11,13\}$ , $C=\{11,13,15\}$ and $D=\{15,17\}$ .

(i) $A\cap B$

\begin{aligned} \textbf{Sol.}\ A\cap B &= \{3,5,7,9,11\}\cap\{7,9,11,13\} \\ &= \{7,9,11\} \end{aligned}

(ii) $B\cap C$

\begin{aligned} \textbf{Sol.}\ B\cap C &= \{7,9,11,13\}\cap\{11,13,15\} \\ &= \{11,13\} \end{aligned}

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

\begin{aligned} \textbf{Sol.}\ A\cap C &= \{3,5,7,9,11\}\cap\{11,13,15\} \\ &= \{11\} \end{aligned}

\begin{aligned} A\cap C\cap D &= \{11\}\cap\{15,17\} \\ &= \phi \end{aligned}

(iv) $A\cap C$

\begin{aligned} \textbf{Sol.}\ A\cap C &= \{3,5,7,9,11\}\cap\{11,13,15\} \\ &= \{11\} \end{aligned}

(v) $B\cap D$

\begin{aligned} \textbf{Sol.}\ B\cap D &= \{7,9,11,13\}\cap\{15,17\} \\ &= \phi \end{aligned}

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

\begin{aligned} \textbf{Sol.}\ B\cup C &= \{7,9,11,13\}\cup\{11,13,15\} \\ &= \{7,9,11,13,15\} \end{aligned}

\begin{aligned} & A\cap(B\cup C) \\ & {=}\;\{3,5,7,9,11\}\cap\{7,9,11,13,15\} \\ & {=}\;\{7,9,11\} \end{aligned}

(vii) $A\cap D$

\begin{aligned} \textbf{Sol.}\ A\cap D &= \{3,5,7,9,11\}\cap\{15,17\} \\ &= \phi \end{aligned}

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

\begin{aligned} \textbf{Sol.}\ B\cup D &= \{7,9,11,13\}\cup\{15,17\} \\ &= \{7,9,11,13,15,17\} \end{aligned}

\begin{aligned} & A\cap(B\cup D) \\ & {=}\;\{3,5,7,9,11\}\cap\{7,9,11,13,15,17\} \\ & {=}\;\{7,9,11\} \end{aligned}

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

\textbf{Sol.}\ A\cap B= \{7,9,11\}

[From (i)]

B\cup C= \{7,9,11,13,15\}

[From (vi)]

\begin{aligned} & (A\cap B)\cap(B\cup C) \\ & {=}\;\{7,9,11\}\cap\{7,9,11,13,15\} \\ & {=}\;\{7,9,11\} \end{aligned}

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

\begin{aligned} \textbf{Sol.}\ A\cup D &= \{3,5,7,9,11\}\cup\{15,17\} \\ &= \{3,5,7,9,11,15,17\} \end{aligned}

B\cup C= \{7,9,11,13,15\}

[From (vi)]

\begin{aligned} & (A\cup D)\cap(B\cup C) \\ & {=}\;\{3,5,7,9,11,15,17\}\cap\{7,9,11,13,15\} \\ & {=}\;\{7,9,11,15\} \end{aligned}

Q7. If $A=\{x:x\text{ is a natural number}\}$ , $B=\{x:x\text{ is an even natural number}\}$ , $C=\{x:x\text{ is an odd natural number}\}$ and $D=\{x:x\text{ is a prime number}\}$ , find

(i) $A\cap B$

(ii) $A\cap C$

(iii) $A\cap D$

(iv) $B\cap C$

(v) $B\cap D$

(vi) $C\cap D$

Sol. In roster form, $A=\{1,2,3,4,\ldots\}$ , $B=\{2,4,6,8,\ldots\}$ , $C=\{1,3,5,7,\ldots\}$ and $D=\{2,3,5,7,11,\ldots\}$ .

(i) Every even natural number is a natural number, so $B\subset A$ .

∴ $A\cap B=B\allowbreak=\{x:x$ is an even natural $\text{number}\}$

(ii) Every odd natural number is a natural number, so $C\subset A$ .

∴ $A\cap C=C\allowbreak=\{x:x$ is an odd natural $\text{number}\}$

(iii) Every prime number is a natural number, so $D\subset A$ .

∴ $A\cap D=D=\{x:x\text{ is a prime number}\}$

(iv) No natural number is both even and odd.

∴ $B\cap C=\phi$

(v) $2$ is the only even prime number.

∴ $B\cap D=\{2\}$

(vi) Every prime number except $2$ is odd.

∴ $C\cap D=\{x:x$ is an odd prime $\text{number}\}$

Q8. Which of the following pairs of sets are disjoint

(Two sets are disjoint if their intersection is $\phi$ .)

(i) $\{1,2,3,4\}$ and $\{x:x$ is a natural number $\text{and }4\le x\le6\}$

Sol. $\{x:x$ is a natural number $\text{and }4\le x\le6\}=\{4,5,6\}$ .

$\{1,2,3,4\}\cap\{4,5,6\}=\{4\}\ne\phi$

∴ The sets are not disjoint.

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

Sol. $\{a,e,i,o,u\}\cap\{c,d,e,f\}=\{e\}\ne\phi$

∴ The sets are not disjoint.

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

Sol. No integer is both even and odd, so the intersection is $\phi$ .

∴ The sets are disjoint.

Q9. 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) $D-C$

Sol.

\begin{aligned} & \textbf{(i)}\ A-B \\ & {=}\;\{3,6,9,12,15,18,21\} -\{4,8,12,16,20\} \\ & {=}\;\{3,6,9,15,18,21\} \end{aligned}

\begin{aligned} & \textbf{(ii)}\ A-C \\ & {=}\;\{3,6,9,12,15,18,21\} -\{2,4,6,8,10,12,14,16\} \\ & {=}\;\{3,9,15,18,21\} \end{aligned}

\begin{aligned} & \textbf{(iii)}\ A-D \\ & {=}\;\{3,6,9,12,15,18,21\}-\{5,10,15,20\} \\ & {=}\;\{3,6,9,12,18,21\} \end{aligned}

\begin{aligned} & \textbf{(iv)}\ B-A \\ & {=}\;\{4,8,12,16,20\} -\{3,6,9,12,15,18,21\} \\ & {=}\;\{4,8,16,20\} \end{aligned}

\begin{aligned} & \textbf{(v)}\ C-A \\ & {=}\;\{2,4,6,8,10,12,14,16\} -\{3,6,9,12,15,18,21\} \\ & {=}\;\{2,4,8,10,14,16\} \end{aligned}

\begin{aligned} & \textbf{(vi)}\ D-A \\ & {=}\;\{5,10,15,20\}-\{3,6,9,12,15,18,21\} \\ & {=}\;\{5,10,20\} \end{aligned}

\begin{aligned} & \textbf{(vii)}\ B-C \\ & {=}\;\{4,8,12,16,20\} -\{2,4,6,8,10,12,14,16\} \\ & {=}\;\{20\} \end{aligned}

\begin{aligned} & \textbf{(viii)}\ B-D \\ & {=}\;\{4,8,12,16,20\}-\{5,10,15,20\} \\ & {=}\;\{4,8,12,16\} \end{aligned}

\begin{aligned} & \textbf{(ix)}\ C-B \\ & {=}\;\{2,4,6,8,10,12,14,16\} -\{4,8,12,16,20\} \\ & {=}\;\{2,6,10,14\} \end{aligned}

\begin{aligned} & \textbf{(x)}\ D-B \\ & {=}\;\{5,10,15,20\}-\{4,8,12,16,20\} \\ & {=}\;\{5,10,15\} \end{aligned}

\begin{aligned} & \textbf{(xi)}\ C-D \\ & {=}\;\{2,4,6,8,10,12,14,16\} -\{5,10,15,20\} \\ & {=}\;\{2,4,6,8,12,14,16\} \end{aligned}

\begin{aligned} & \textbf{(xii)}\ D-C \\ & {=}\;\{5,10,15,20\} -\{2,4,6,8,10,12,14,16\} \\ & {=}\;\{5,15,20\} \end{aligned}

Q10. If $X=\{a,b,c,d\}$ and $Y=\{f,b,d,g\}$ , find

(i) $X-Y$

(ii) $Y-X$

(iii) $X\cap Y$

Sol. Given, $X=\{a,b,c,d\}$ and $Y=\{f,b,d,g\}$ .

\textbf{(i)}\ X-Y=\{a,b,c,d\}-\{f,b,d,g\}\allowbreak=\{a,c\}

\textbf{(ii)}\ Y-X=\{f,b,d,g\}-\{a,b,c,d\}\allowbreak=\{f,g\}

\textbf{(iii)}\ X\cap Y=\{a,b,c,d\}\cap\{f,b,d,g\}\allowbreak=\{b,d\}

Q11. If $R$ is the set of real numbers and $Q$ is the set of rational numbers, then what is $R-Q$ ?

Sol. $R-Q$ contains all real numbers which are not rational numbers.

∴ $R-Q$ is the set of irrational numbers.

Q12. State whether each of the following statement is true or false. Justify your answer.

(Two sets are disjoint if their intersection is $\phi$ .)

(i) $\{2,3,4,5\}$ and $\{3,6\}$ are disjoint sets.

Sol. $\{2,3,4,5\}\cap\{3,6\}=\{3\}\ne\phi$

∴ The statement is false.

(ii) $\{a,e,i,o,u\}$ and $\{a,b,c,d\}$ are disjoint sets.

Sol. $\{a,e,i,o,u\}\cap\{a,b,c,d\}=\{a\}\ne\phi$

∴ The statement is false.

(iii) $\{2,6,10,14\}$ and $\{3,7,11,15\}$ are disjoint sets.

Sol. $\{2,6,10,14\}\cap\{3,7,11,15\}=\phi$

∴ The statement is true.

(iv) $\{2,6,10\}$ and $\{3,7,11\}$ are disjoint sets.

Sol. $\{2,6,10\}\cap\{3,7,11\}=\phi$

∴ The statement is true.