NCERT Solutions Class 12

NCERT Solutions Class 12

Relations and Functions : NCERT Solutions – Class 12 Maths (Ex 4)

Exercise 1.4

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 .


 

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.


 

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

 

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


 

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

 

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 


 

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 

* 1 2 3 4 5
1 1 2 3 4 5
2 2 2 6 4 10
3 3 x 3 12 15
4 4 4 12 4 20
5 5 x 15 20 5

Here, 2 * 3 = 6 A

Therefore, the operation * is not a binary operation.


 

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.


 

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.


 

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.


 

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.


 

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.


 

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.

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