NCERT Solutions for Class 11 Maths Chapter 1 Sets Ex 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

Opting for the NCERT solutions for Ex 1.4 Class 11 Maths is considered as the best option for the CBSE students when it comes to exam preparation. This chapter consists of many exercises. Out of which we have provided the Exercise 1.4 Class 11 Maths NCERT solutions on this page in PDF format. You can download this solution as per your convenience or you can study it directly from our website/ app online.

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