Exercise 1.4
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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 .
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.
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 |
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
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 |
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 
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
| * | 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.
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.
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.
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.
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.
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.
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.