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Edited By Ramraj Saini | Updated on Dec 03, 2023 01:35 PM IST | #CBSE Class 12th

**NCERT Solutions for Exercise 1.4 Class 12 Maths Chapter 1 Relations and Functions **are discussed here. These NCERT solutions are created by subject matter expert at Careers360 considering the **latest syllabus and pattern of CBSE 2023-24**. 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.

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**12th class Maths exercise 1.4 **answers** **are designed as per the students demand covering comprehensive, step by step solutions of every problem. Practice these questions and answers to command the concepts, boost confidence and in depth understanding of concepts. Students can find all exercise together using the link provided below.

- Relations and Functions Exercise 1.1
- Relations and Functions Exercise 1.2
- Relations and Functions Exercise 1.3
- Relations and Functions Miscellaneous Exercise

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(i) On , define ∗ by

** Answer: **

(i) On , define ∗ by

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

(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.

(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.

(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.

(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

** 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.

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.

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Just Study 40% Syllabus and Score upto 100%

Download EBook**Comprehensive Coverage:**The solutions encompass all the topics covered in ex 1.4 class 12, ensuring a thorough understanding of the concepts.**Step-by-Step Solutions:**In this class 12 maths ex 1.4, each problem is solved systematically, providing a stepwise approach to aid in better comprehension for students.**Accuracy and Clarity:**Solutions for class 12 ex 1.4 are presented accurately and concisely, using simple language to help students grasp the concepts easily.**Conceptual Clarity:**In this 12th class maths exercise 1.4 answers, emphasis is placed on conceptual clarity, providing explanations that assist students in understanding the underlying principles behind each problem.**Inclusive Approach:**Solutions for ex 1.4 class 12 cater to different learning styles and abilities, ensuring that students of various levels can grasp the concepts effectively.**Relevance to Curriculum:**The solutions for class 12 maths ex 1.4 align closely with the NCERT curriculum, ensuring that students are prepared in line with the prescribed syllabus.

NCERT Solutions Subject Wise

- NCERT solutions class 12 chemistry
- NCERT solutions for class 12 physics
- NCERT solutions for class 12 biology
- NCERT solutions for class 12 mathematics

Subject wise NCERT Exemplar solutions

- NCERT Exemplar Class 12th Maths
- NCERT Exemplar Class 12th Physics
- NCERT Exemplar Class 12th Chemistry
- NCERT Exemplar Class 12th Biology

Happy learning!!!

1. Which concepts are covered in Exercise 1.4 Class 12 Maths?

These concepts are discussed in class 12 ex 1.4. Concepts like proving a binary operation associative, commutative etc. are mentioned in the Exercise 1.4 Class 12 Maths. Practice 12th class maths exercise 1.4 answers to command the cocepts.

2. What are the important topics in chapter relations and functions ?

Definitions of relations and functions, types of relations, types of functions, composition of functions, invertible function and binary operations are the important topics in this chapter.

3. How much weightage is held by the chapter relations and functions for CBSE board exam ?

The weightage of chapter relation and function is more than 5% in the CBSE board examination.

4. How are the NCERT solutions helpful in the board exam ?

As students can assess from previous year questions that many questions are asked directly from NCERT exercise. Hence to score well in the examination, it is required to practice NCERT exercise for Class 12 Maths before the exam.

5. What are relations in Class 12 Maths?

These concepts are discussed in class 12 maths ex 1.4. In maths, relation defines the relationship between sets of values of ordered pairs. Practice these questions to get deeper understanding.

6. Mention the number of questions in Exercise 1.4 Class 12 Maths ?

13 questions are the in Exercise 1.4 Class 12 Maths

7. How many exercises are there in the NCERT Class 12 Maths chapter 1 ?

There are a total of 5 exercises including a miscellaneous exercise in the NCERT Class 12 Maths chapter 1.

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6 minHave a question related to CBSE Class 12th ?

You can use them people also used problem

Hi,

The Medhavi National Scholarship Program, under the Human Resources & Development Mission (HRDM), offers financial assistance to meritorious students through a scholarship exam. To be eligible, candidates must be between 16 and 40 years old as of the last date of registration and have at least passed the 10th grade from a recognized board. Higher qualifications, such as 11th/12th grade, graduation, post-graduation, or a diploma, are also acceptable.

To apply, download the Medhavi App from the Google Play Store, sign up, and read the detailed notification about the scholarship exam. Complete the registration within the app, take the exam from home using the app, and receive your results within two days. Following this, upload the necessary documents and bank account details for verification. Upon successful verification, the scholarship amount will be directly transferred to your bank account.

The scholarships are categorized based on the marks obtained in the exam: Type A for those scoring 60% or above, Type B for scores between 50% and 60%, and Type C for scores between 40% and 50%. The cash scholarships range from Rs. 2,000 to Rs. 18,000 per month, depending on the exam and the marks obtained.

Since you already have a 12th-grade qualification with 84%, you meet the eligibility criteria and can apply for the Medhavi Scholarship exam. Preparing well for the exam can increase your chances of receiving a higher scholarship.

hello mahima,

If you have uploaded screenshot of your 12th board result taken from CBSE official website,there won,t be a problem with that.If the screenshot that you have uploaded is clear and legible. It should display your name, roll number, marks obtained, and any other relevant details in a readable forma.ALSO, the screenshot clearly show it is from the official CBSE results portal.

hope this helps.

Hello Akash,

If you are looking for important questions of class 12th then I would like to suggest you to go with previous year questions of that particular board. You can go with last 5-10 years of PYQs so and after going through all the questions you will have a clear idea about the type and level of questions that are being asked and it will help you to boost your class 12th board preparation.

You can get the Previous Year Questions (PYQs) on the official website of the respective board.

I hope this answer helps you. If you have more queries then feel free to share your questions with us we will be happy to assist you.

**
Thank you and wishing you all the best for your bright future.
**

Hello student,

**
If you are planning to appear again for class 12th board exam with PCMB as a private candidate here is the right information you need:
**

- No school admission needed! Register directly with CBSE. (But if you want to attend the school then you can take admission in any private school of your choice but it will be waste of money)
- You have to appear for the 2025 12th board exams.
- Registration for class 12th board exam starts around September 2024 (check CBSE website for exact dates).
- Aim to register before late October to avoid extra fees.
- Schools might not offer classes for private students, so focus on self-study or coaching.

**
Remember
**
, these are tentative dates based on last year. Keep an eye on the CBSE website ( https://www.cbse.gov.in/ ) for the accurate and official announcement.

I hope this answer helps you. If you have more queries then feel free to share your questions with us, we will be happy to help you.

**
Good luck with your studies!
**

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