Exercise 1.4 (Revised) - Chapter 1 -Sets - Ncert Solutions class 11 - Maths

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Chapter 1 - Sets | NCERT Solutions for Class 11 Maths

Ex 1.4 Question 1.

Find the union of each of the following pairs of sets:
(i) $X=\{1,3,5\}$ and $Y=\{1,2,3\}$
(ii) $\mathrm{A}=\{\mathrm{a}, \mathrm{e}, \mathrm{i}, \mathrm{o}, \mathrm{u}\}$ and $\mathrm{B}=\{\mathrm{a}, \mathrm{b}, \mathrm{c}\}$
(iii) $A=\{x: x$ is a natural number and multiple of 3$\}$ and $B=\{x: x$ is a natural number less than 6$\}$
(iv) $\mathrm{A}=\{\mathrm{x}: \mathrm{x}$ is a natural number and $1<x \leq 6\}$ and $\mathrm{B}=\{\mathrm{x}$ : $\mathrm{x}$ is a natural number and $6<x<10\}$
(v) $A=\{1,2,3\}$ and $B=\emptyset$

Answer.

(i) $\mathrm{X} \cup \mathrm{Y}=\{1,2,3,5\}$
(ii) $\mathrm{A} \cup \mathrm{B}=\{a, b, c, e, i, o, u\}$
(iii) $\mathrm{A} \cup \mathrm{B}=\{1,2,3,4,5,6,9,12,15, \ldots \ldots \ldots .$.$\} or$
$A \cup B=\{x: x=1,2,4,5$ or a multiple of 3$\}$
(iv) $\mathrm{A} \cup \mathrm{B}=\{2,3,4,5,6,7,8,9\}$ or
$A \cup B=\{x: 1<x<10, x \in N\}$
(v) $\mathrm{A} \cup \mathrm{B}=\{1,2,3\}$

Ex 1.4 Question 2.

Let $\mathrm{A}=\{a, b\}$ and $\mathrm{B}=\{a, b, c\}$. Is $\mathrm{A} \subset \mathrm{B}$ ? What is $\mathrm{A} \cup \mathrm{B}$ ? Ans. Given: $\mathrm{A}=\{a, b\}$ and $\mathrm{B}=\{a, b, c\}$.

Here all elements of set A are present in set B.
$\therefore \mathrm{A} \subset \mathrm{B}$ and
$\mathrm{A} \cup \mathrm{B}=\{a, b, c\}=\mathrm{B}$
Ex 1.4 Question 3.

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

Answer.

Given: A and B are two sets such that $\mathrm{A} \subset \mathrm{B}$

Taking $\mathrm{A}=\{1,2\}$ and $B=\{1,2,3\}$, then $A \cup B=\{1,2,3\}=B$

So $\mathrm{A} \cup \mathrm{B}=\mathrm{B}$
Ex 1.4 Question 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) $\mathrm{A} \cup \mathrm{B}$
(ii) $\mathrm{A} \cup \mathrm{C}$
(iii) $B \cup C$
(iv) $B \cup D$
(v) $\mathrm{A} \cup \mathrm{B} \cup \mathrm{C}$
(vi) $A \cup B \cup D$

(vii) $B \cup C \cup D$

Answer.

Given: $\mathrm{A}=\{1,2,3,4\}, \mathrm{B}=\{3,4,5,6\}, \mathrm{C}$ $=\{5,6,7,8\}$ and $D=\{7,8,9,10\}$
(i) $\mathrm{A} \cup \mathrm{B}=\{1,2,3,4\} \cup\{3,4,5,6\}$

So, $A \cup B=\{1,2,3,4,5,6\}$
(ii) $\mathrm{A} \cup \mathrm{C}=\{1,2,3,4\} \cup\{5,6,7,8\}$

So, $A \cup C=\{1,2,3,4,5,6,7,8\}$
(iii) $B \cup C=\{3,4,5,6\} \cup\{5,6,7,8\}$
So, $B \cup C=\{3,4,5,6,7,8\}$
(iv) $\mathrm{B} \cup \mathrm{D}=\{3,4,5,6\} \cup\{7,8,9,10\}$
So, $B \cup D=\{3,4,5,6,7,8,9,10\}$
(v) $\mathrm{A} \cup \mathrm{B} \cup \mathrm{C}=\{1,2,3,4\} \cup\{3,4,5,6\} \cup\{5,6,7,8\}$
So, $A \cup B \cup C=\{1,2,3,4,5,6,7,8\}$
(vi) $\mathrm{A} \cup \mathrm{B} \cup \mathrm{D}=\{1,2,3,4\} \cup\{3,4,5,6\} \cup\{7,8,9,10\}$
So, $A \cup B \cup D=\{1,2,3,4,5,6,7,8,9,10\}$
(vii) $B \cup C \cup D=\{3,4,5,6\} \cup\{5,6,7,8\} \cup\{7,8,9,10\}$
So, $B \cup C \cup D=\{3,4,5,6,7,8,9,10\}$
Ex 1.4 Question 5.

Find the intersections of each pair of sets of question 1 above.

Answer.

(i) $\mathrm{X} \cap \mathrm{Y}=\{1,3\}$
(ii) $\mathrm{A} \cap \mathrm{B}=\{a\}$3

(iii) $\mathrm{A} \cap \mathrm{B}=\{3\}$
(iv) $\mathrm{A} \cap \mathrm{B}=\phi$
(v) $\mathrm{A} \cap \mathrm{B}=\emptyset$
Ex 1.4 Question 6.

If $A=\{3,5,7,9,11\}, B=\{7,9,11,13\}, C=\{11,13,15\}$ and $D=\{15,17\}$; find:
(i) $\mathrm{A} \cap \mathrm{B}$
(ii) $\mathrm{B} \cap \mathrm{C}$

(iii) $\mathrm{A} \cap \mathrm{C} \cap \mathrm{D}$
(iv) $\mathrm{A} \cap \mathrm{C}$
(v) $\mathrm{B} \cap \mathrm{D}$
(vi) $\mathrm{A} \cap(\mathrm{B} \cup \mathrm{C})$
(vii) $\mathrm{A} \cap \mathrm{D}$
(viii) $\mathrm{A} \cap(B \cup \mathrm{D})$
(ix) $(\mathrm{A} \cap \mathrm{B}) \cap(\mathrm{B} \cup \mathrm{C})$
(x) $(\mathrm{A} \cup \mathrm{D}) \cap(B \cup C)$

Answer.

Given: $\mathrm{A}=\{3,5,7,9,11\}$,
$B=\{7,9,11,13\}$,
$C=\{11,13,15\}$ and $D=\{15,17\}$
(i) $\mathrm{A} \cap \mathrm{B}=\{3,5,7,9,11\} \cap\{7,9,11,13\}$
$
=\{7,9,11\}
$
(ii) $\mathrm{B} \cap \mathrm{C}=\{7,9,11,13\} \cap\{11,13,15\}$
$
=\{11,13\}
$

(iii) $\mathrm{A} \cap \mathrm{C} \cap \mathrm{D}$
$
=\{3,5,7,9,11\} \cup\{11,13,15\} \cap\{15,17\}=\varnothing
$
(iv) $\mathrm{A} \cap \mathrm{C}=\{3,5,7,9,11\} \cap\{11,13,15\}=\{11\}$
(v) $\mathrm{B} \sim \mathrm{D}=\{7,9,11,13\} \cap\{15,17\}=\phi$
(vi) $A \cap(B \cup C)=\{3,5,7,9,11\} \cap(\{7,9,11,13\} \cup\{11,13,15\})$ $=\{3,5,7,9,11\} \cap\{7,9,11,13,15,17\}=\{7,9,11\}$

$
\begin{aligned}
& \text { (vii) } \mathrm{A} \cap \mathrm{C}=\{3,5,7,9,11\} \cap\{15,17\}=\phi \\
& \text { (viii) } \mathrm{A} \cap(\mathrm{B} \cup \mathrm{D})=\{3,5,7,9,11\} \cap(\{7,9,11,13\} \cup\{15,17\}) \\
& =\{3,5,7,9,11\} \cap\{7,9,11,13,15,17\}=\{7,9,11\} \\
& \text { (ix) }(\mathrm{A} \cap \mathrm{B}) \cap(\mathrm{B} \cup \mathrm{C}) \\
& =(\{3,5,7,9,11\} \cap\{7,9,11,13\}) \cap(\{7,9,11,13\} \cup\{11,13,15\}) \\
& =\{7,9,11\} \cap\{7,9,11,13,15\}=\{7,9,11\} \\
& \text { (x) }(\mathrm{A} \cup \mathrm{D}) \cap(\mathrm{B} \cup \mathrm{C}) \\
& =(\{3,5,7,9,11\} \cup\{15,17\}) \cap(\{7,9,11,13\} \cup\{11,13,15\}) \\
& =\{3,5,7,9,11,15,17\} \cap\{7,9,11,13,15\}=\{7,9,11,15\}
\end{aligned}
$
Ex 1.4 Question 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) $\mathrm{A} \cap \mathrm{C}$
(iii) $\mathrm{A} \sim \mathrm{D}$
(iv) $\mathrm{B} C$
(v) $B \cap D$
(vi) $\mathrm{C} \mathrm{D}$

Answer.

(i) $\mathrm{A} \Omega \mathrm{B}=\{x: x$ is a natural number $\} \cap\{x: x$ is an even natural number $\}=\mathrm{B}$
(ii) A $\cap=\{x: x$ is a natural number $\} \cap\{x: x$ is an odd natural number $\}=\mathrm{C}$
(iii) $\mathrm{A} \cap \mathrm{D}=\{x: x$ is a natural number $\} \cap\{x: x$ is a prime number $\}=\mathrm{D}$

(iv) $\mathrm{B} \cap \mathrm{C}=\{x: x$ is an even natural number $\} \cap\{x: x$ is an odd natural number $\}=\phi$
(v) $\mathrm{B} \cap \mathrm{D}=\mathrm{B} \cap \mathrm{C}=\{x: x$ is an even natural number $\} \cap\{x: x$ is a prime number $\}=\{2\}$
(vi) $\mathrm{C} \cap \mathrm{D}=\{x: x$ is an odd natural number $\} \cap\{x: x$ is a prime number $\}$ $=\{x: x$ is an odd prime number $\}$

Ex 1.4 Question 8.

Which of the following pair 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 $\}$

Answer.

(i) Let $\mathrm{A}=\{1,2,3,4\}$ and $\mathrm{B}=\{x: x$ is a natural number and $4 \leq x \leq 6\}=\{4,5,6\}$
$\therefore \mathrm{A} \cap \mathrm{B}=\{4\}$
Therefore, A and B are not disjoint.
(ii) Let $\mathrm{A}=\{a, e, i, o, u\}$ and $\mathrm{B}=\{c, d, e, f\}$
$\therefore \mathrm{A} \cap \mathrm{B}=\{\mathrm{e}\}$
Therefore, $\mathrm{A}$ and $\mathrm{B}$ are not disjoint.
(iii) Let $\mathrm{A}=\{x ; x$ is an even integer $\}$ and $\mathrm{B}=\{x ; x$ is an odd integer $\}$
$\therefore \mathrm{A} \cap \mathrm{B}=\phi$
Therefore, A and B are disjoint.
Ex 1.4 Question 9.

If $\mathrm{A}=\{3,6,9,12,15,18,21\}, B=\{4,8,12,16,20\}, C=\{2,4,6,8,10,12,14,16\}$,
$D=\{5,10,15,20\}$; find:

$\text { (i) } \mathrm{A}-\mathrm{B}$

(ii) $\mathrm{A}-\mathrm{C}$
(iii) A - D
(iv) $\mathrm{B}-\mathrm{A}$
(v) $\mathrm{C}-\mathrm{A}$
(vi) $\mathrm{D}-\mathrm{A}$
(vii) B-C
(viii) B - D
(ix) $\mathrm{C}-\mathrm{B}$
(x) $\mathrm{D}-\mathrm{B}$
(xi) $\mathrm{C}-\mathrm{D}$
(xii) D-C

Answer.

Given: $\mathrm{A}=\{3,6,9,12,15,18,21\}$,
$
\begin{aligned}
& B=\{4,8,12,16,20\}, \\
& C=\{2,4,6,8,10,12,14,16\}, \\
& D=\{5,10,15,20\} ;
\end{aligned}
$
$
\begin{aligned}
& \text { (i) } \mathrm{A}-\mathrm{B}=\{3,6,9,12,15,18,21\}-\{4,8,12,16,20\} \\
& =\{3,6,9,15,18,21\}
\end{aligned}
$
$
\begin{aligned}
& \text { (ii) } \mathrm{A}-\mathrm{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}
& \text { (iii) } \mathrm{A}-\mathrm{D}=\{3,6,9,12,15,18,21\}-\{5,10,15,20\} \\
& =\{3,6,9,12,18,21\}
\end{aligned}
$
$
\begin{aligned}
& \text { (iv) } \mathrm{B}-\mathrm{A}=\{4,8,12,16,20\}-\{3,6,9,12,15,18,21\} \\
& =\{4,8,16,20\}
\end{aligned}
$
(v) $\mathrm{C}-\mathrm{A}=\{2,4,6,8,10,12,14,16\}-\{3,6,9,12,15,18,21\}$
$
=\{2,4,8,10,14,16\}
$

$
\begin{aligned}
& \text { (vi) } D-A=\{5,10,15,20\}-\{3,6,9,12,15,18,21\} \\
& =\{5,10,20\} \\
& \text { (vii) } B-C=\{4,8,12,16,20\}-\{2,4,6,8,10,12,14,16\}=\{20\} \\
& \text { (viii) } B-D=\{4,8,12,16,20\}-\{5,10,15,20\} \\
& =\{4,8,12,16\} \\
& \text { (ix) } C-B=\{2,4,6,8,10,12,14,16\}-\{4,8,12,16,20\} \\
& =\{2,6,10,14\} \\
& \text { (x) } D-B=\{5,10,15,20\}-\{4,8,12,16,20\} \\
& =\{5,10,15\} \\
& \text { (xi) } C-D=\{2,4,6,8,10,12,14,16\}-\{5,10,15,20\} \\
& =\{2,4,6,8,12,14,16\} \\
& \text { (xii) } D-C=\{5,10,15,20\}-\{2,4,6,8,10,12,14,16\} \\
& =\{5,15,20\}
\end{aligned}
$
Ex 1.4 Question 10.

If $\mathbf{X}=\{a, b, c, d\}$ and $\mathbf{Y}=\{f, b, d, g\}$, find:
(i) $X-Y$

(ii) $\mathrm{Y}-\mathrm{X}$
(iii) $\mathrm{X} \cap \mathrm{Y}$

Answer.

Given: $\mathrm{X}=\{a, b, c, d\}$ and $\mathrm{Y}=\{f, b, d, g\}$
(i) $\mathrm{X}-\mathrm{Y}=\{a, b, c, d\}-\{f, b, d, g\}=\{a, c\}$
(ii) $\mathrm{Y}-\mathrm{X}=\{f, b, d, g\}-\{a, b, c, d\}=\{f, g\}$
(iii) $\mathrm{X} \cap \mathrm{Y}=\{a, b, c, d\} \cap\{f, b, d, g\}=\{b, d\}$

Ex 1.4 Question 11.

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

Answer.

We know that set of real numbers contain rational and irrational numbers.
Therefore, $\mathrm{R}-\mathrm{Q}=$ set of irrational numbers.
Ex 1.4 Question 12.

State whether each of the following statements 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.

Answer.

(i) Let $A=\{2,3,4,5\}$ and $B=\{3,6\}$
$\therefore \mathrm{A} \cap \mathrm{B}=\{3\}$
$\therefore$ A and B are not disjoint. Therefore, statement is false.
(ii) Let $\mathrm{A}=\{a, e, i, o, u\}$ and $\mathrm{B}=\{a, b, c, d\}$
$\therefore \mathrm{A} \cap \mathrm{B}=\{a\}$
$\therefore$ A and B are not disjoint. Therefore, statement is false.
(iii) Let $A=\{2,6,10,14\}$ and $B=\{3,7,11,15\}$
$\therefore \mathrm{A} \cap \mathrm{B}=\phi$
$\therefore \mathrm{A}$ and $\mathrm{B}$ are disjoint. Therefore, statement is true.

(iv) Let $\mathrm{A}=\{2,6,10\}$ and $\mathrm{B}=\{3,7,11\}$
$
\therefore \mathrm{A} \cap \mathrm{B}=\emptyset
$
$\therefore \mathrm{A}$ and $\mathrm{B}$ are disjoint. Therefore, statement is true.