NCERT Solutions class 12 Maths Exercise 1.4

NCERT Solutions class 12 Maths Exercise 1.4 Class 12 Maths book solutions are available in PDF format for free download. These ncert book chapter wise questions and answers are very helpful for CBSE board exam. CBSE recommends NCERT books and most of the questions in CBSE exam are asked from NCERT text books. Class 12 Maths chapter wise NCERT solution for Maths part 1 and Maths part 2 for all the chapters can be downloaded from our website and myCBSEguide mobile app for free.

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NCERT Solutions class 12 Maths Relations and Functions 

1. Determine whether or not each of the definition of * given below gives a binary operation. In the event that * is not a binary operation, given justification for this.

(i) On  define * by  

(ii) On  define * by

(iii) On R, define * by

(iv) On  define * by

(v) On  define * by

Ans. (i) On  = {1, 2, 3, …..}, 

Let    

Therefore, operation * is not a binary operation on .

(ii) On  = {1, 2, 3, …..},

Let     

Therefore, operation * is a binary operation on .

(iii) on R (set of real numbers)

Let    R

Therefore, operation * is a binary operation on R.

(iv) On  = {1, 2, 3, …..},

Let    

Therefore, operation * is a binary operation on .

(v) On  = {1, 2, 3, …..},

Let    

Therefore, operation * is a binary operation on .

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NCERT Solutions class 12 Maths Exercise 1.4

2. For each binary operation * defined below, determine whether * is commutative or associative:

(i) On  define

(ii) On Q, define

(iii) On Q, define

(iv) On  define

(v) On  define

(vi) On R – {– 1}, define

Ans. (i) For commutativity:  and  =

For associativity:  =

Also,  =

 

Therefore, the operation * is neither commutative nor associative.

(ii) For commutativity:  and

For associativity:  =

Also,  =

 

Therefore, the operation * is commutative but not associative.

(iii) For commutativity:  and  =

For associativity:  =

Also,  =

 

Therefore, the operation * is commutative and associative.

(iv) For commutativity:  and

For associativity:  =

Also,  =

 

Therefore, the operation * is commutative but not associative.

(v) For commutativity:  and

 

For associativity:  =

Also,  =

 

Therefore, the operation * is neither commutative nor associative.

(vi) For commutativity:  and    

For associativity:  =

Also,  =

 

Therefore, the operation * is neither commutative nor associative.

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NCERT Solutions class 12 Maths Exercise 1.4

3. Consider the binary operation  on the set {1, 2, 3, 4, 5} defined by Write the operation table of the operation  

Ans.  Let A = {1, 2, 3, 4, 5} defined by  i.e., minimum of  and

	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

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NCERT Solutions class 12 Maths Exercise 1.4

4. Consider a binary operation * on the set {1, 2, 3, 4, 5} given by the following multiplication table (table 1.2).

(i) Compute (2 * 3) * 4 and 2 * (3 * 4)

(ii) Is * commutative?

(iii) Compute (2 * 3) * (4 * 5)

(Hint: Use the following table)

Table 1.2

*	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

Ans. (i) 2 * 3 = 1 and 3 * 4 = 1

Now (2 * 3) * 4 = 1 * 4 = 1 and 2 * (3 * 4) = 2 * 1 = 1

(ii) 2 * 3 = 1 and 3 * 4 = 1

2 * 3 = 3 * 2 and other element of the given set.

Hence the operation is commutative.

(iii) (2 * 3) * (4 * 5) = 1 * 1 = 1

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NCERT Solutions class 12 Maths Exercise 1.4

5. Let *’ be the binary operation on the set {1, 2, 3, 4, 5} defined by  H.C.F. of  and  Is the operation *’ same as the operation * defined in Exercise 4 above? Justify your answer.

Ans. Let A = {1, 2, 3, 4, 5} and  H.C.F. of  and

*’	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

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NCERT Solutions class 12 Maths Exercise 1.4

6. Let * be the binary operation on N given by  L.C.M. of  and   Find:

(i) 5 * 7,  10 * 16

(ii) Is * commutative?

(iii) Is * associative?

(iv) Find the identity of * in N.

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

Ans.  L.C.M. of  and

(i) 5 * 7 = L.C.M. of 5 and 7 = 35

20 * 16 = L.C.M. of 20 and 16 = 80

(ii)  L.C.M. of  and  = L.C.M. of  and  =

Therefore, operation * is commutative.

(iii)  =

=

Similarly,

Thus,

Therefore, the operation is associative.

(iv) Identity of * in N = 1 because  = L.C.M. of  and 1 =

(v) Only the element 1 in N is invertible for the operation * because

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NCERT Solutions class 12 Maths Exercise 1.4

7. Is * defined on the set {1, 2, 3, 4, 5} by  L.C.M. of  and  a binary operation? Justify your answer.

Ans. Let A = {1, 2, 3, 4, 5} and  L.C.M. of  and

Here, 2 * 3 = 6 A

Therefore, the operation * is not a binary operation.

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NCERT Solutions class 12 Maths Exercise 1.4

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

Ans.  H.C.F. of  and

(i)  H.C.F. of  and  = H.C.F. of  and  =

Therefore, operation * is commutative.

(ii)  =

=

Therefore, the operation is associative.

Therefore, there does not exist any identity element.

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NCERT Solutions class 12 Maths Exercise 1.4

9. Let * be a binary operation on the set Q of rational numbers as follows:

(i)      (ii)   

(iii)      (iv)  

(v)      (vi)  

Find which of the binary operations are commutative and which are associative.

Ans. (i)    operation * is not commutative.

And

Here,    operation * is not associative.

(ii)   operation * is commutative.

And

Here,    operation * is not associative.

(iii)  and

Therefore, operation * is not commutative.

And

Here,   operation * is not associative.

(iv)   operation * is commutative.

And

Here,    operation * is not associative.

(v)   operation * is commutative.

 And

Here,    operation * is associative.

(vi)  and    operation * is not commutative.

And

Here,   operation * is not associative.

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NCERT Solutions class 12 Maths Exercise 1.4

10. Show that none of the operations given above the identity.

Ans. Let the identity be I.

(i)

(ii)

(iii)

(iv)

(v)

(vi)

Therefore, none of the operations given above has identity.

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NCERT Solutions class 12 Maths Exercise 1.4

11. Let A = N x N and * be the binary operation on A defined by  

Show that * is commutative and associative. Find the identity element for * on A, if any.

Ans. A = N x N and * is a binary operation defined on A.

  The operation is commutative

Again, 

And 

Here,    The operation is associative.

Let identity function be , then 

For identity function  

And for

As 0 N, therefore, identity-element does not exist.

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NCERT Solutions class 12 Maths Exercise 1.4

12. State whether the following statements are true or false. Justify:

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

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

Ans. (i) * being a binary operation on N, is defined as

Hence operation * is not defined, therefore, the given statement is false.

(ii) * being a binary operation on N.

    

Thus,   , therefore the given statement is true.

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NCERT Solutions class 12 Maths Exercise 1.4

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?

Ans.     The operation * is commutative.

Again, 

And 

 The operation * is not associative.

Therefore, option (B) is correct.

NCERT Solutions class 12 Maths Exercise 1.4

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