# NCERT Solutions for Exercise 1.4 Class 12 Maths Chapter 1 - Relations and Functions

NCERT Solutions for class 12 maths chapter 1 exercise 1.4 talks about commutative, associative, binary operations etc. Exercise 1.4 Class 12 Maths has questions which includes finding whether a relation follows binary operation or not. After analysis from previous year questions it is clear that NCERT Solutions for class 12 maths chapter 1 exercise 1.4 plays an important role to score well in CBSE class 12 board exam. Also it has a significant contribution in competitive exams like JEE main. Hence it is recommended for students to solve all the questions of Class 12th maths chapter 1 exercise 1.4 to score well in their examination. The NCERT chapter Relations and Functions also consists the following exercise for practice.

Relations and Functions Exercise 1.1

Relations and Functions Exercise 1.2

Relations and Functions Exercise 1.3

Relations and Functions Miscellaneous Exercise

## ** NCERT Solutions for Class 12 Maths Chapter 1 Relations and Functions: Exercise 1.4 **

** Question:1(i) ** Determine whether or not each of the definition of ∗ given below gives a binary operation. In the event that ∗ is not a binary operation, give justification for this.

(i) On , define ∗ by

** Answer: **

(i) On , define ∗ by

It is not a binary operation as the image of under * is .

** Question:1(ii) ** Determine whether or not each of the definition of ∗ given below gives a binary operation. In the event that ∗ is not a binary operation, give justification for this.

(ii) On , define ∗ by

** Answer: **

(ii) On , define ∗ by

We can observe that for ,there is a unique element ab in .

This means * carries each pair to a unique element in .

Therefore,* is a binary operation.

** Question:1(iii) ** Determine whether or not each of the definition of ∗ given below gives a binary operation. In the event that ∗ is not a binary operation, give justification for this.

(iii) On , define ∗ by

** Answer: **

(iii) On , define ∗ by

We can observe that for ,there is a unique element in .

This means * carries each pair to a unique element in .

Therefore,* is a binary operation.

** Question:1(iv) ** Determine whether or not each of the definition of ∗ given below gives a binary operation. In the event that ∗ is not a binary operation, give justification for this.

(iv) On , define ∗ by

** Answer: **

(iv) On , define ∗ by

We can observe that for ,there is a unique element in .

This means * carries each pair to a unique element in .

Therefore,* is a binary operation.

** Question:1(v) ** Determine whether or not each of the definition of ∗ given below gives a binary operation. In the event that ∗ is not a binary operation, give justification for this

(v) On , define ∗ by

** Answer: **

(v) On , define ∗ by

* carries each pair to a unique element in .

Therefore,* is a binary operation.

** Question:2(i) ** For each operation ∗ defined below, determine whether ∗ is binary, commutative or associative.

(i)On , define

** Answer: **

a*b=a-b

b*a=b-a

so * is not commutative

(a*b)*c=(a-b)-c

a*(b*c)=a-(b-c)=a-b+c

(a*b)*c not equal to a*(b*c), so * is not associative

** Question:2(ii) ** For each operation ∗ defined below, determine whether ∗ is binary, commutative or associative.

(ii) On , define

** Answer: **

(ii) On , define

ab = ba for all

ab+1 = ba + 1 for all

for

where

operation * is not associative.

** Question:2(iii) ** For each operation ∗ defined below, determine whether ∗ is binary, commutative or associative.

(iii) On , define

** Answer: **

(iii) On , define

ab = ba for all

for all

for

operation * is commutative.

operation * is associative.

** Question:2(iv) ** For each operation ∗ defined below, determine whether ∗ is binary, commutative or associative.

(iv) On , define

** Answer: **

(iv) On , define

ab = ba for all

2ab = 2ba for all

for

the operation is commutative.

where

operation * is not associative.

** Question:2(v) ** For each operation ∗ defined below, determine whether ∗ is binary, commutative or associative.

(v) On , define

** Answer: **

(v) On , define

and

for

the operation is not commutative.

where

operation * is not associative.

** Question:2(vi) ** For each operation ∗ defined below, determine whether ∗ is binary, commutative or associative.

(vi) On , define

** Answer: **

(iv) On , define

and

for

the operation is not commutative.

where

operation * is not associative.

** Question:3 ** Consider the binary operation on the set defined by . Write the operation table of the operation .

** Answer: **

for

The operation table of the operation is given by :

| 1 | 2 | 3 | 4 | 5 |

1 | 1 | 1 | 1 | 1 | 1 |

2 | 1 | 2 | 2 | 2 | 2 |

3 | 1 | 2 | 3 | 3 | 3 |

4 | 1 | 2 | 3 | 4 | 4 |

5 | 1 | 2 | 3 | 4 | 5 |

** Question:4(i) ** Consider a binary operation ∗ on the set given by the following multiplication table (Table 1.2).

(i) Compute and

(Hint: use the following table)

** Answer: **

(i)

** Question:4(ii) ** Consider a binary operation ∗ on the set given by the following multiplication table (Table 1.2).

(ii) Is ∗ commutative?

(Hint: use the following table)

** Answer: **

(ii)

For every , we have . Hence it is commutative.

** Question:4(iii) ** Consider a binary operation ∗ on the set { given by the following multiplication table (Table 1.2).

(iii) Compute (2 ∗ 3) ∗ (4 ∗ 5).

(Hint: use the following table)

** Answer: **

(iii) (2 ∗ 3) ∗ (4 ∗ 5).

from the above table

** Question:5 ** Let ∗′ be the binary operation on the set defined by . Is the operation ∗′ same as the operation ∗ defined

in Exercise 4 above? Justify your answer.

** Answer: **

for

The operation table is as shown below:

| 1 | 2 | 3 | 4 | 5 |

1 | 1 | 1 | 1 | 1 | 1 |

2 | 1 | 2 | 1 | 2 | 1 |

3 | 1 | 1 | 3 | 1 | 1 |

4 | 1 | 2 | 1 | 4 | 1 |

5 | 1 | 1 | 1 | 1 | 5 |

The operation ∗′ same as the operation ∗ defined in Exercise 4 above.

** Question:6(i) ** let ∗ be the binary operation on N given by . Find

(i) 5 ∗ 7, 20 ∗ 16

** Answer: **

a*b=LCM of a and b

(i) 5 ∗ 7, 20 ∗ 16

** Question:6(ii) ** Let ∗ be the binary operation on N given by . Find

(ii) Is ∗ commutative?

** Answer: **

(ii) for all

Hence, it is commutative.

** Question:6(iii) ** Let ∗ be the binary operation on N given by a ∗ b = L.C.M. of a and b. Find

(iii) Is ∗ associative?

** Answer: **

a b = L.C.M. of a and b

(iii)

Hence, the operation is associative.

** Question:6(iv) ** Let ∗ be the binary operation on N given by . Find

(iv) the identity of ∗ in N

** Answer: **

(iv) the identity of ∗ in N

We know that

for

Hence, 1 is the identity of ∗ in N.

** Question 6(v) ** Let ∗ be the binary operation on N given by a ∗ b = L.C.M. of a and b. Find

(v) Which elements of N are invertible for the operation ∗?

** Answer: **

An element a is invertible in N

if

Here a is inverse of b.

a*b=1=b*a

a*b=L.C.M. od a and b

a=b=1

So 1 is the only invertible element of N

** Question:7 ** Is ∗ defined on the set by a binary operation? Justify your answer.

** Answer: **

A =

Operation table is as shown below:

| 1 | 2 | 3 | 4 | 5 |

1 | 1 | 2 | 3 | 4 | 5 |

2 | 2 | 2 | 6 | 4 | 10 |

3 | 3 | 6 | 3 | 12 | 15 |

4 | 4 | 4 | 12 | 4 | 20 |

5 | 5 | 10 | 15 | 20 | 5 |

From the table, we can observe that

Hence, the operation is not a binary operation.

** Question:8 ** Let ∗ be the binary operation on N defined by a ∗ b = H.C.F. of a and b. Is ∗ commutative? Is ∗ associative? Does there exist identity for this binary

operation on N?

Answer:

a ∗ b = H.C.F. of a and b for all

H.C.F. of a and b = H.C.F of b and a for all

Hence, operation ∗ is commutative.

For ,

Hence, ∗ is associative.

An element will be identity for operation * if for .

Hence, the operation * does not have any identity in N.

** Question:9(i) ** Let ∗ be a binary operation on the set Q of rational numbers as follows:

(i) Find which of the binary operations are commutative and which are associative.

** Answer: **

On the set Q ,the operation * is defined as .It is observed that:

here

Hence, the * operation is not commutative.

It can be observed that

for all

The operation * is not associative.

** Question:9(ii) ** Let ∗ be a binary operation on the set Q of rational numbers as follows:

(ii) Find which of the binary operations are commutative and which are associative.

** Answer: **

On the set Q ,the operation * is defines as .It is observed that:

For

Hence, the * operation is commutative.

It can be observed that

for all

The operation * is not associative.

** Question:9(iii) ** Let ∗ be a binary operation on the set Q of rational numbers as follows:

(iii) Find which of the binary operations are commutative and which are associative.

** Answer: **

On the set Q ,the operation * is defines as .It is observed that:

For

for

Hence, the * operation is not commutative.

It can be observed that

for all

The operation * is not associative.

** Question:9(iv) ** Let ∗ be a binary operation on the set Q of rational numbers as follows:

(iv) Find which of the binary operations are commutative and which are associative.

** Answer: **

On the set Q ,the operation * is defined as .It is observed that:

For

for

Hence, the * operation is commutative.

It can be observed that

for all

The operation * is not associative.

** Question:9(v) ** Let ∗ be a binary operation on the set Q of rational numbers as follows:

(v) Find which of the binary operations are commutative and which are associative.

** Answer: **

On the set Q ,the operation * is defines as .It is observed that:

For

for

Hence, the * operation is commutative.

It can be observed that

for all

The operation * is associative.

** Question:9(vi) ** Let ∗ be a binary operation on the set Q of rational numbers as follows:

(vi) Find which of the binary operations are commutative and which are associative.

** Answer: **

On the set Q ,the operation * is defines as .It is observed that:

For

for

Hence, the * operation is not commutative.

It can be observed that

for all

The operation * is not associative.

** Question:10 ** Find which of the operations given above has identity.

** Answer: **

An element will be identity element for operation *

if for all

when .

Hence, has identity as 4.

However, there is no such element which satisfies above condition for all rest five operations.

Hence, only (v) operations have identity.

** Question:11 ** Let and ∗ be the binary operation on A defined by Show that ∗ is commutative and associative. Find the identity element for ∗ on A, if any.

** Answer: **

and ∗ be the binary operation on A defined by

Let

Then,

We have

Thus it is commutative.

Let

Then,

Thus, it is associative.

Let will be a element for operation * if for all .

i.e.

This is not possible for any element in A .

Hence, it does not have any identity.

** Question:12(i) ** State whether the following statements are true or false. Justify.

(i) For an arbitrary binary operation ∗ on a set N,

** Answer: **

(i) For an arbitrary binary operation ∗ on a set N,

An operation * on a set N as

Then , for b=a=2

Hence, statement (i) is false.

** Question:12(ii) ** State whether the following statements are true or false. Justify.

(ii) If ∗ is a commutative binary operation on N, then

** Answer: **

(ii) If ∗ is a commutative binary operation on N, then

R.H.S

(* is commutative)

( as * is commutative)

= L.H.S

Hence, statement (ii) is true.

** Question:13 ** Consider a binary operation ∗ on N defined as . Choose the correct answer.

(A) Is ∗ both associative and commutative?

(B) Is ∗ commutative but not associative?

(C) Is ∗ associative but not commutative?

(D) Is ∗ neither commutative nor associative?

** Answer: **

A binary operation ∗ on N defined as .

For

Thus, it is commutative.

where

Hence, it is not associative.

Hence, B is the correct option.

**More About NCERT Solutions for Class 12 Maths Chapter 1 Exercise 1.4**

The NCERT Class 12 Maths chapter Relations and Functions has a total of 5 exercises including miscellaneous exercise. **Exercise 1.4 Class 12 Maths** covers solutions to 13 main questions and their sub-questions. Most of the questions are related to concepts of binary operations which includes commutative, associative operations etc. Hence NCERT Solutions for Class 12 Maths chapter 1 exercise 1.4 is recommended for learning these concepts to score well in the exam.

Also Read| NCERT Notes For Class 12 Mathematics Chapter 1

**Benefits of NCERT Solutions for Class 12 Maths Chapter 1 Exercise 1.4**

The Class 12 Maths NCERT syllabus chapter 1.4 exercise is provided above in a comprehensive manner which is solved by subject matter experts .

Students are recommended to practice Exercise 1.4 Class 12 Maths to prepare for topics like binary operations which includes commutative, associative operations etc.

These

**Class 12 Maths chapter 1 exercise 1.4**solutions can be referred by students to revise just before the exam.NCERT syllabus for Class 12 Maths chapter 1 exercise 1.4 can be used to prepare similar topics of physics also.

### Also see-

NCERT exemplar solutions class 12 maths chapter 1

NCERT solutions for class 12 maths chapter 1

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NCERT solutions for class 12 mathematics

Subject wise NCERT Exemplar solutions

NCERT Exemplar Class 12th Maths

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