NCERT Solutions for Class 12 Maths Exercise 1.3 Chapter 1 Relations and Functions – FREE PDF Download

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 $a * b = 5.2 {(3)}^{2} = 36.8 \in$ R

Therefore, operation * is a binary operation on R.

(iv) On = {1, 2, 3, …..},

Let $a * b = | 3 - 7 | = | - 4 | = 4 \in$

Therefore, operation * is a binary operation on .

(v) On = {1, 2, 3, …..},

Let $a * b = 5 \times 7 = 35 \in$

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: = $\frac{a}{\frac{b}{c + 1} + 1} = \frac{a (c + 1)}{b + c + 1}$

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

We observe that the operation *’ is the same as the operation * in Exp.4.

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

(i) 5 * 7, 20 * 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?

(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 Ch 1 Relations and Functions

NCERT Solutions for Class 12 Maths Exercise 1.3 Chapter 1 Relations and Functions – FREE PDF Download

NCERT Solutions class 12 Maths Relations and Functions

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

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

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

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.

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

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

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?

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

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

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

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

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

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.