# NCERT Solutions For Class 11 Maths Chapter 1 Exercise 1.4

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## Chapter 1 Ex.1.4 Question 1

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

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

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

(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 $$1 \lt x \le 6$$ }

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

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

### Solution

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

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

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

$$A\cup B=\left\{ a,~b,~c,~e,~i,~o,~u \right\}$$

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

$${\rm{A = }}\left\{ {3,6,9 \ldots } \right\}$$

$$B =$$ {$$x:x$$ is a natural number less than $$6$$}

$$B = \left\{ {1,2,3,4,5,6} \right\}$$

$$A \cup B = \left\{ {1,2,4,5,3,6,9,12 \ldots } \right\}$$

Therefore, $$A \cup B =$$ { $$x:x = 1,2,4,5$$ or a multiple of 3}

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

$$A = \left\{ {2,3,4,5,6} \right\}$$

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

$$B = \left\{ {7,8,9} \right\}$$

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

Therefore, $$A \cup B = \left\{ {x:x \in N\;\;{\rm{and}}\;\;1 \lt x \lt 10} \right\}$$

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

$$A \cup B = \left\{ {1,2,3} \right\}$$

## Chapter 1 Ex.1.4 Question 2

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

### Solution

Here, $$A=\left\{ a,~b \right\}$$ and $$B=\left\{ a,~b,~c \right\}$$

We see, $$B$$ consists all the elements of $$A$$

Hence, Yes $$A \subset B$$.

$A\cup B=\left\{ a,~b,~c \right\}$

## Chapter 1 Ex.1.4 Question 3

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

### Solution

If $$A$$ and $$B$$ are two sets such that $$A \subset B$$, that means B consists all the elements of $$A$$

Then, $$A \cup B = B$$.

## Chapter 1 Ex.1.4 Question 4

If $$A = \left\{ {1,2,3,4} \right\},\;\;B = \left\{ {3,4,5,6} \right\},\;\;C = \left\{ {5,6,7,8} \right\}$$ and $$D = \left\{ {7,8,9,10} \right\}$$; 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) $$B \cup C \cup D$$

### Solution

$$A = \left\{ {1,2,3,4} \right\},\;\;B = \left\{ {3,4,5,6} \right\},\;\;C = \left\{ {5,6,7,8} \right\}$$ and $$D = \left\{ {7,8,9,10} \right\}$$

(i) $$A \cup B = \left\{ {1,2,3,4,5,6} \right\}$$

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

(iii) $$B \cup C = \left\{ {3,4,5,6,7,8} \right\}$$

(iv) $$B \cup D = \left\{ {3,4,5,6,7,8,9,10} \right\}$$

(v) $$A \cup B \cup C = \left\{ {1,2,3,4,5,6,7,8} \right\}$$

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

(vii) $$B \cup C \cup D = \left\{ {3,4,5,6,7,8,9,10} \right\}$$

## Chapter 1 Ex.1.4 Question 5

Find the intersection of each pair of sets for the following.

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

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

(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 $$1 \lt x \le 6$$}

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

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

### Solution

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

$X \cap Y = \left\{ {1,3} \right\}$

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

$A \cap B = \left\{ a \right\}$

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

$$B =$$ {$$x:x$$ is a natural number less than $$6$$}

$A \cap B = \left\{ 3 \right\}$

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

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

$A \cap B = \phi$

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

$A \cap B = \phi$

## Chapter 1 Ex.1.4 Question 6

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

(i) $$A \cap B$$

(ii) $$B \cap C$$

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

(iv) $$A \cap C$$

(v) $$B \cap D$$

(vi) $$A \cap \left( {B \cup C} \right)$$

(vii) $$A \cap D$$

(viii) $$A \cap \left( {B \cup D} \right)$$

(ix) $$\left( {A \cap B} \right) \cap \left( {B \cup C} \right)$$

(x) $$\left( {A \cup D} \right) \cap \left( {B \cup C} \right)$$

### Solution

(i) $$A \cap B = \left\{ {7,9,11} \right\}$$

(ii) $$B \cap C = \left\{ {11,13} \right\}$$

(iii) $$A \cap C \cap D = \left\{ {A \cap C} \right\} \cap D = \left\{ {11} \right\} \cap \left\{ {15,17} \right\} = \phi$$

(iv) $$A \cap C = \left\{ {11} \right\}$$

(v) $$B \cap D = \phi$$

(vi) $$A \cap \left( {B \cup C} \right) = \left( {A \cap B} \right) \cup \left( {A \cap C} \right) = \left\{ {7,9,11} \right\} \cup \left\{ {11} \right\} = \left\{ {7,9,11} \right\}$$

(vii) $$A \cap D = \phi$$

(viii) $$A\cap \left( B\cup D \right)=\left( A\cap B \right)\cup \left( A\cap D \right)=\left\{ 7,9,11 \right\}~\cup \phi =\left\{ 7,9,11 \right\}$$

(ix) $$\left( {A \cap B} \right) \cap \left( {B \cup C} \right) = \left\{ {7,9,11} \right\} \cap \left\{ {7,9,11,13,15} \right\} = \left\{ {7,9,11} \right\}$$

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

## Chapter 1 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) $$A \cap C$$

(iii) $$A \cap D$$

(iv) $$B \cap C$$

(v) $$B \cap D$$

(vi) $$C \cap D$$

### Solution

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

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

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

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

(i) $$A \cap B =$$ {$$x:x$$ is a even natural number} $$= B$$

(ii) $$A \cap C =$$ {$$x:x$$ is an odd natural number} $$= C$$

(iii) $$A \cap D =$$ {$$x:x$$ is a prime number} $$= D$$

(iv) $$B \cap C = \phi$$

(v) $$B \cap D = \left\{ 2 \right\}$$

(vi) $$C \cap D =$${$$x:x$$ is an odd prime number}

## Chapter 1 Ex.1.4 Question 8

Which of the following pairs of sets are disjoint

(i) $$\left\{ {1,2,3,4} \right\}$$ and {$$x:x$$ is a natural number and $$4 \le x \le 6$$}

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

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

### Solution

(i) $$\left\{ {1,2,3,4} \right\}$$

{$$x:x$$ is a natural number and $$4 \le x \le 6$$}$$= \left\{ {4,5,6} \right\}$$

Now, $$\left\{ {1,2,3,4} \right\} \cap \left\{ {4,5,6} \right\} = \left\{ 4 \right\}$$

Therefore, this pair of sets is not disjoint.

(ii) $$\left\{ a,~e,~i,~o,~u \right\}\cap \left\{ c,~d,~e,~f \right\}=\left\{ e \right\}$$

Therefore, this pair of sets 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.

## Chapter 1 Ex.1.4 Question 9

If $$A = \left\{ {3,6,9,12,15,18,21} \right\},$$ $$B = \left\{ {4,8,12,16,20} \right\}$$, $$C = \left\{ {2,4,6,8,10,12,14,16} \right\},$$ $$D = \left\{ {5,10,15,20} \right\};$$ 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) $$D - C$$

### Solution

(i) $$A - B = \left\{ {3,6,9,15,18,21} \right\}$$

(ii) $$A-C = \left\{ {3,9,15,18,21} \right\}$$

(iii) $$A-D = \left\{ {3,6,9,12,18,21} \right\}$$

(iv) $$B - A = \left\{ {4,8,16,20} \right\}$$

(v) $$C-A = \left\{ {2,4,8,10,14,16} \right\}$$

(vi) $$D - A = \left\{ {5,10,20} \right\}$$

(vii) $$B-C = \left\{ {20} \right\}$$

(viii) $$B - D = \left\{ {4,8,12,16} \right\}$$

(ix) $$C - B = \left\{ {2,6,10,14} \right\}$$

(x) $$D-B = \left\{ {5,10,15} \right\}$$

(xi) $$C - D = \left\{ {2,4,6,8,12,14,16} \right\}$$

(xii) $$D - C = \left\{ {5,15,20} \right\}$$

## Chapter 1 Ex.1.4 Question 10

If $$X=\left\{ a,~b,~c,~d \right\}$$ and $$Y=\left\{ f,~b,~d,g \right\}$$ , find

(i) $$X - Y$$

(ii) $$Y-X$$

(iii) $$X \cap Y$$

### Solution

(i) $$X-Y=\left\{ a,~c \right\}$$

(ii) $$Y-X = \left\{ {f,g} \right\}$$

(iii) $$X\cap Y=\left\{ b,~d \right\}$$

## Chapter 1 Ex.1.4 Question 11

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

### Solution

$$R:$$ set of real numbers

$$Q:$$ set of rational numbers

Therefore, $$R – Q$$ is a set of irrational numbers.

## Chapter 1 Ex.1.4 Question 12

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

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

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

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

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

### Solution

(i) False

Since, $$3 \in \left\{ {2,3,4,5} \right\}$$ and $$3 \in \left\{ {3,6} \right\}$$

Therefore, $$\left\{ {2,3,4,5} \right\} \cap \left\{ {3,6} \right\} = \left\{ 3 \right\}$$

(ii) False

Since, $$a \in \left\{ {a,e,i,o,u} \right\}$$and $$a \in \left\{ {a,b,c,d} \right\}$$

Therefore, $$\left\{ {a,e,i,o,u} \right\} \cap \left\{ {a,b,c,d} \right\} = \left\{ a \right\}$$

(iii) True

Since, $$\left\{ {2,6,10,14} \right\} \cap \left\{ {3,7,11,15} \right\} = \phi$$

(iv) True

Since, $$\left\{ {2,6,10} \right\} \cap \left\{ {3,7,11} \right\} = \phi$$

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