NCERT Solutions for Class 11 Maths Chapter 1: Sets - Exercise 1.4

Exercise 1.4

1. Find the union of each of the following pairs of sets.

(i)\[{\mathbf{x}} = \{ {\mathbf{1}},{\mathbf{3}},{\mathbf{5}}\} {\text{ }}{\mathbf{Y}} = \{ {\mathbf{1}},{\mathbf{2}},{\mathbf{3}}\} \]

Ans: $X = \{ 1,3,5\} Y = \{ 1,2,3\} $

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

(ii) \[{\mathbf{A}} = {\text{ }}\left\{ {{\mathbf{a}},{\text{ }}{\mathbf{e}},{\text{ }}{\mathbf{i}},{\text{ }}{\mathbf{o}},{\text{ }}{\mathbf{u}}} \right\}{\text{ }}{\mathbf{B}} = {\text{ }}\left\{ {{\mathbf{a}},{\text{ }}{\mathbf{b}},{\text{ }}{\mathbf{c}}} \right\}\]

Ans: $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$\} $

Ans: $A = \{ x:x$ 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 \leqslant 6\}  = \{ 2,3,4,5,6\} $

$B = \{ x:x$ is a natural number and $6 < x < 10\}  = \{ 7,8,9\} $

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

$\therefore A \cup B = \{ x:x \in N$ and $1 < x < 10\} $

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

Ans: $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 $A \cup B$ ?

Ans: 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 $A \cup B$?

Ans: If ${\text{A}}$ and ${\text{B}}$ 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) $A \cup B$

Ans: $A \cup B = \{ 1,2,3,4,5,6\} $

(ii) $A \cup C$

Ans: $A \cup C = \{ 1,2,3,4,5,6,7,8\} $

(iii) $B \cup C$

Ans: $B \cup C = \{ 3,4,5,6,7,8\} $

(iv) $B \cup D$

Ans: $B \cup D = \{ 3,4,5,6,7,8,9,10\} $

(v) \[A \cup B \cup C\]

Ans: $A \cup B \cup C = \{ 1,2,3,4,5,6,7,8\} $

(vi) \[A \cup B \cup D\]

Ans: $A \cup B \cup D = \{ 1,2,3,4,5,6,7,8,9,10\} $

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

Ans: $B \cup C \cup D = \{ 3,4,5,6,7,8,9,10\} $

5. Find the intersection of each pair of sets.

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

Ans: $X \cap Y = \{ 1,3\} $

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

Ans: $A \cap B = \{ a\} $

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

$B = \{ x:x$ is a natural number less than \[{\mathbf{6}}\]$\} $

Ans: $\therefore A \cap B = \{ 3\} $

(iv) $A = \{ x:x$ is a natural number and $1 < x \leqslant 6\} $

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

Ans: $A = \{ x:x$ is a natural number and $1 < x \leqslant 6\}  = \{ 2,3,4,5,6\} $

$B = \{ x:x$ is a natural number and $6 < x < 10\}  = \{ 7,8,9\} $

$A \cap B = \emptyset $

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

Ans: $A \cap B = \emptyset $

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

(i) $A \cap B$

Ans: $A \cap B = \{ 7,9,11\} $

(ii) $B \cap C$

Ans: $B \cap C = \{ 11,13\} $

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

Ans: $A \cap C \cap D = \{ A \cap C\}  \cap D = \{ 11\}  \cap \{ 15,17\}  = \emptyset $

(iv) $A \cap C$

Ans: $A \cap C\{ 11\} $

(v) $B \cap D$

Ans: $B \cap D = \emptyset $

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

Ans: $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$

$ = \{ 7,9,11\}  \cup \{ 11\}  = \{ 7,9,11\} $

(vii) $A \cap D$

Ans: $A \cap D = \emptyset $

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

Ans: $A \cap (B \cup D) = (A \cap B) \cup (A \cap D)$

$ = \{ 7,9,11\}  \cup \emptyset  = \{ 7,9,11\} $

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

Ans: $(A \cap B) \cap (B \cup C) = \{ 7,9,11\}  \cap \{ 7,9,11,13,15\}  = \{ 7,9,11\} $

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

Ans: $(A \cup D) \cap (B \cup C) = \{ 3,5,7,9,11,15,17\}  \cap \{ 7,9,11,13,15\} $

$ = \{ 7,9,11,15\} $

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

$A = \{ x:x$ is a natural number $\}  = \{ 1,2,3,4,5 \ldots \} $

$B = \{ x:x$ is an even natural number $\}  = \{ 2,4,6,8 \ldots \} $

$C = \{ x:x$ is an odd natural number $\}  = \{ 1,3,5,7,9 \ldots \} $

$D = \{ x:x$ is a prime number $\}  = \{ 2,3,5,7 \ldots \} $

(i) $A \cap B$

Ans: $A \cap B = \{ x:x$ is an even natural number $\}  = B$

(ii) $A \cap C$

Ans: $A \cap C = \{ x:x$ is an odd natural number $\}  = C$

(iii) $A \cap D$

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

(iv) $B \cap C$

Ans: $B \cap C = \emptyset $

(v) $B \cap D$

Ans: $B \cap D = \{ 2\} $

(vi) $C \cap D$

Ans: $C \cap D = \{ x:x$ is odd prime number $\} $

8. Which of the following pairs of sets are disjoint

(i) $\{ 1,2,3,4\} $ and $\{ x:x$ is a natural number and \[4 \leqslant x \leqslant 6\} \]

Ans: $\{ 1,2,3,4\} $

$\{ x:x$ is a natural number and $4 \leqslant x \leqslant 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} and {c,d,e,f}

Ans: $\{ 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} and $\{ x:x$ is an odd integer}

Ans: $\{ x:x$ is an even integer $\}  \cap \{ x:x$ is an odd integer $\}  = \emptyset $

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

Ans: $A - B = \{ 3,6,9,15,18,21\} $

(ii) A-C

Ans: $A - C = \{ 3,9,15,18,21\} $

(iii) A-D

Ans: $A - D = \{ 3,6,9,12,18,21\} $

(iv) B-A

Ans: $B - A = \{ 4,8,16,20\} $

(v) C-A

Ans: $C - A = \{ 2,4,8,10,14,16\} $

(vi) D-A

Ans:$D - A = \{ 5,10,20\} $

(viii) B-C

Ans: $B - C = \{ 20\} $

(viii) B-D

Ans: $B - D = \{ 4,8,12,16\} $

(ix) C-B

Ans: $C - B = \{ 2,6,10,14\} $

(x) D-B

Ans: $D - B = \{ 5,10,15\} $

(xi) C-D

Ans:  $C - D = \{ 2,4,6,8,12,14,16\} $

(xii) D-C

Ans: $D - C = \{ 5,15,20\} $

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

(i) X-Y

Ans: $X - Y = \{ a,c\} $

(ii) Y-X

Ans: $Y - X = \{ f,g\} $

(iii) $X \cap Y$

Ans:$X \cap Y = \{ b,d\} $

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

Ans: R. Set of real numbers

Q. Set of rational numbers

Therefore, ${\text{R}} - {\text{Q}}$ is a set of irrational number.

12. State whether each of the following statement is true or false. Justify you Ans:

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

Ans: False

As $3 \in \{ 2,3,4,5\} ,3 \in \{ 3,6\} $

$ \Rightarrow \{ 2,3,4,5\}  \cap \{ 3,6\}  = \{ 3\} $

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

Ans: 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) {2, 6, 10, 14} and {3, 7, 11, 15} are disjoint sets.

Ans: True

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

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

Ans:  True

As $\{ 2,6,10\}  \cap \{ 3,7,11\}  = \emptyset $

NCERT Solutions for Class 11 Maths Chapter 1 Sets Exercise 1.4

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