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# NCERT Solutions for Class 12 Maths Relations and Functions

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## NCERT Solutions for Class 12 Maths Relations and Functions Exercise 1.4

NCERT Solutions for Class 12 Maths Relations and Functions Exercise 1.4: Ques No 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, give justification for this.

(i) On Z+, define by a b = a – b

NCERT Solutions:

Given that ∗ defined by a ∗ b = a – b in Z+.

Since the image of (1, 2) under * is 1 * 2 = 1 − 2 = −1 ∉ Z+, the operation * is not a binary operation.

(ii) On Z+, define by a b = ab

NCERT Solutions:

Given that ∗ defined by a ∗ b = ab in Z+.

Since for each a, b ∈ Z+, * carries each pair (a, b) to a unique element a * b = ab in Z+, then the operation * is a binary operation.

(iii) On Z+, define by a b = ab2

NCERT Solutions:

Given that ∗ defined by a ∗ b = ab2 in R.

Since for each a, b ∈ R, * carries each pair (a, b) to a unique element a * b = ab2 in R, then the operation * is a binary operation.

(iv) On Z+, define by a b = |a – b|

NCERT Solutions:

Given that ∗ defined by a ∗ b = | a – b | in Z+.

Since for each a, b ∈ Z+, * carries each pair (a, b) to a unique element a * b = | a – b | in Z+, then the operation * is a binary operation.

(v) On Z+, define by a b = a

NCERT Solutions:

Given that ∗ defined by a ∗ b = a in Z+.

Since for each a, b ∈ Z+, * carries each pair (a, b) to a unique element a * b = a in Z+, then the operation * is a binary operation.

NCERT Solutions for Class 12 Maths Relations and Functions Exercise 1.4: Ques No 1. For each binary operation ∗ defined below, determine whether ∗ is commutative or associative.

(i) On Z+, define by a b = a – b

NCERT Solutions:

Given that ∗ defined by a ∗ b = a – b in Z.

Let a = 2, b = 3 in Z. Then,

2 * 3 = 2 – 3 = – 1 and 3 * 2 = 3 – 2 = 1

We get that 2 * 3 ≠ 3 * 2

Thus, the given operation * is not commutative.

Let a = 2, b = 3, c = 4 in Z. Then,

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

2 * (3 * 4) = 2 * (3 – 4) = 2 * –1 = 2 – (–1) = 2 + 1 = 3

We get that (2 * 3) * 4 ≠ 2 * (3 * 4)

Thus, the given operation * is not associative.

(ii) On Z+, define by a b = ab + 1

NCERT Solutions:

Given that ∗ defined by a ∗ b = ab + 1 in Q.

Let a = 2, b = 3 in Q. Then,

2 * 3 = 2×3 + 1 = 6 + 1 = 7 and 3 * 2 = 3×2 + 1 = 6 + 1 = 7

We get that 2 * 3 = 3 * 2

Thus, the given operation * is commutative.

Let a = 2, b = 3, c = 4 in Q. Then,

(2 * 3) * 4 = (2×3 + 1) * 4 = 7 * 4 = 7×4 + 1 = 29

2 * (3 * 4) = 2 * (3×4 + 1) = 2 * 13 = 2×13 + 1 = 27

We get that (2 * 3) * 4 ≠ 2 * (3 * 4)

Thus, the given operation * is not associative.

(iii) On Z+, define by a b = ab/2

NCERT Solutions:

Given that ∗ defined by a ∗ b = ab/2 in Q.

Let a = 2, b = 3 in Q. Then, 2 * 3 = (2 × 3)/2 = 3 and 3 * 2 = (3 × 2) / 2 = 3

We get that 2 * 3 = 3 * 2. Thus, the given operation * is commutative.

Let a = 2, b = 3, c = 4 in Q. Then, We get that (2 * 3) * 4 = 2 * (3 * 4). Thus, the given operation * is associative.

(iv) On Z+, define by a b = 2ab

NCERT Solutions:

Given that ∗ defined by a ∗ b = 2ab in Z+.

Let a = 2, b = 3 in Z. Then, 2 * 3 = 2(2×3) = 26 = 64 and 3 * 2 = 2(3×2) = 26 = 64

We get that 2 * 3 = 3 * 2. Thus, the given operation * is commutative.

Let a = 2, b = 3, c = 4 in Z+. Then,

(2 * 3) * 4 = 2(2×3) * 4 = 64 * 4 = 2(64×4) = 2256

2 * (3 * 4) = 2 * 2(3×4)  = 2 * 4096 = 2(2×4096) = 28192

We get that (2 * 3) * 4 ≠ 2 * (3 * 4)

Thus, the given operation * is not associative.

(v) On Z+, define by a b = ab

NCERT Solutions:

Given that ∗ defined by a ∗ b = ab in Z+.

Let a = 2, b = 3 in Z. Then, 2 * 3 = 23 = 8 and 3 * 2 = 32 = 9

We get that 2 * 3 ≠ 3 * 2. Thus, the given operation * is not commutative.

Let a = 2, b = 3, c = 4 in Z+. Then,

(2 * 3) * 4 = 23 * 4 = 8 * 4 = 84 = 4096 = 212

2 * (3 * 4) = 2 * 34  = 2 * 81 = 281

We get that (2 * 3) * 4 ≠ 2 * (3 * 4). Thus, the given operation * is not associative.

(vi) On Z+, define by a b = a/(b + 1)

NCERT Solutions:

Given that ∗ defined by a ∗ b = a/(b + 1) in R – {–1}.

Let a = 2, b = 3 in R – {–1}. Then, 2 * 3 = 2/(3 + 1) = ½ and 3 * 2 = 3/(2 + 1) = 1

We get that 2 * 3 ≠ 3 * 2. Thus, the given operation * is not commutative.

Let a = 2, b = 3, c = 4 in R – {–1}. Then, NCERT Solutions for Class 12 Maths Relations and Functions Exercise 1.4: Ques No 3. Consider the binary operation ∧ on the set {1, 2, 3, 4, 5} defined by a ∧ b = min {a, b}. Write the operation table of the operation ∧.

NCERT Solutions:

Given that the binary operation ∧ on the set {1, 2, 3, 4, 5} defined by a ∧ b = min {a, b}.

Thus, the operation table for the given operation is as below. NCERT Solutions for Class 12 Maths Relations and Functions Exercise 1.4: Ques No 4. Consider the binary operation ∧ on the set {1, 2, 3, 4, 5} is given by following multiplication table. (i) Compute (2 3) 4 and 2 (3 4)

NCERT Solutions:

From the given table, we have 2 * 3 = 1, 1 * 4 = 1, 3 * 4 = 1 and 2 * 1 = 1. Thus, (2 * 3) = 1 * 4 = 1 and 2 * (3 * 4) = 2 * 1 = 1.

(ii) Is * commutative?

NCERT Solutions:

For every a, b ∈ {1, 2, 3, 4, 5}, we have a * b = b * a. Therefore, the operation * is commutative.

(iii) Compute (2 ∗ 3) ∗ (4 ∗ 5).

NCERT Solutions:

From the given table, we have 2 * 3 = 1, 4 * 5 = 1, and 1 * 1 = 1. Thus, (2 * 3) * (4 * 5) = 1 * 1 = 1.

NCERT Solutions for Class 12 Maths Relations and Functions Exercise 1.4: Ques No 5. Let ∗′ be the binary operation on the set {1, 2, 3, 4, 5} defined by a ∗′ b = H.C.F. of a and b. Is the operation ∗′ same as the operation ∗ defined in Exercise 4 above? Justify your answer.

NCERT Solutions:

Given that the binary operation ∗′ on the set {1, 2, 3, 4, 5} defined by a ∗′ b = HCF of a and b.

Thus, the operation table for the given operation ∗′ is as below. In the question no 4, the operation table for the given operation ∗ is as below. We observe that the operation tables for the operations * and *′ are the same. Thus, the operation *′ is same as the operation*.

NCERT Solutions for Class 12 Maths Relations and Functions Exercise 1.4: Ques No 6. Let ∗ be the binary operation on N given by a ∗ b = L.C.M. of a and b.

(i) Find 5 7, 20 16

NCERT Solutions:

Given that the binary operation on N given by a ∗ b = L.C.M. of a and b.

Thus, 5 * 7 = LCM of 5 and 7 = 35 and 20 * 16 = LCM of 20 and 16 = 80

(ii) Is * commutative?

NCERT Solutions:

Given that the binary operation on N given by a ∗ b = L.C.M. of a and b

Since LCM of (a, b) and LCM of (b, a) is same for all N, then a ∗ b = b * a

Thus, the given operation * is commutative.

(iii) Is * associative?

NCERT Solutions:

Given that the binary operation on N given by a ∗ b = L.C.M. of a and b

We have

(a * b) * c = (L.C.M of a and b) * c = LCM of a, b, and c

a * (b * c) = a * (LCM of b and c) = L.C.M of a, b, and c

Thus, (a * b) * c = a * (b * c).

Hence, the given operation * is associative.

(iv) Find the identity of * in N.

NCERT Solutions:

Given that the binary operation on N given by a ∗ b = L.C.M. of a and b

Since LCM of (a, 1) and LCM of (1, a) is same for all N and that is 1,

then a ∗ 1 = 1 * a = LCM of a and 1 = a

Thus, 1 is the identity of the given operation *.

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

NCERT Solutions:

Given that the binary operation on N given by a ∗ b = L.C.M. of a and b

An element a in N is invertible with respect to the operation * if there exists an element b in N, such that a * b = c = b * a.

Thus, we have c = 1

This means that L.C.M of a and b = 1 = L.C.M of b and a. This case is possible only when a and b are equal to 1.

Thus, 1 is the only invertible element of N with respect to the operation *.

NCERT Solutions for Class 12 Maths Relations and Functions Exercise 1.4: Ques No 7. Is ∗ defined on the set {1, 2, 3, 4, 5} by a ∗ b = L.C.M. of a and b a binary operation? Justify your answer.

NCERT Solutions:

Given that ∗ defined on the set {1, 2, 3, 4, 5} by a ∗ b = L.C.M. of a and b

Thus, the operation table for the given operation ∗ is as below. From the table, we observe that 6, 10, 12, 15, 20 ∉ A. Thus, the given operation * is not a binary operation.

NCERT Solutions for Class 12 Maths Relations and Functions Exercise 1.4: Ques No 8. Let ∗ be the binary operation on N defined by a ∗ b = H.C.F. of a and b. Is ∗ commutative? Is ∗ associative? Does there exist identity for this binary operation on N?

NCERT Solutions:

Given that ∗ is the binary operation on N defined by a ∗ b = H.C.F. of a and b.

We know that H.C.F. of a and b = H.C.F. of b and a for all a, b ∈ N.

Thus, we get a * b = b * a. Hence, the given operation * is commutative.

For a, b, c ∈ N, we have

(a * b) * c = (H.C.F. of a and b) * c = H.C.F. of a, b and c

a *(b * c) = a *(H.C.F. of b and c) = H.C.F. of a, b, and c

Thus, we get (a * b) * c = a * (b * c). Hence, the given operation * is associative.

Now, an element e ∈ N will be the identity for the operation * if a * e = a = e* a for all a ∈ N. But this relation is not true for any a ∈ N.

Thus, the given operation * does not have any identity in N.

NCERT Solutions for Class 12 Maths Relations and Functions Exercise 1.4: Ques No 9. Let ∗ be a binary operation on the set Q of rational numbers as follows: Find which of the binary operations are commutative and which are associative.

(i) a * b = a – b

NCERT Solutions:

Given that ∗ be a binary operation as a ∗ b = a – b

Since a – b ≠ b – a, then a * b ≠ b * a. Thus, the given operation a ∗ b = a – b is not commutative.

For a, b, c, we have, (a * b) * c = (a – b) * c = (a – b) – c = a – b – c

a * (b * c) = a * (b – c) = a – (b – c) = a – b + c

Since a – b – c ≠ a – b + c, then (a * b) * c ≠ a * (b * c).

Thus, the given operation a ∗ b = a – b is not associative.

(ii) a * b = a2 + b2

NCERT Solutions:

Given that ∗ be a binary operation as a ∗ b = a2 + b2

Since a2 + b2 = b2 + a2, then a * b = b * a

Thus, the given operation a ∗ b = a2 + b2 is commutative.

For a, b, c, we have

(a * b) * c = (a2 + b2) * c = (a2 + b2)2 + c2

a * (b * c) = a * (b2 + c2) = a2 + (b2 + c2)2

Since (a2 + b2)2 + c2 ≠ a2 + (b2 + c2)2, then (a * b) * c ≠ a * (b * c).

Thus, the given operation a ∗ b = a2 + b2 is not associative.

(iii) a * b = a + ab

NCERT Solutions:

Given that ∗ be a binary operation as a ∗ b = a + ab

Since a + ab ≠ b + ba, then a * b ≠ b * a

Thus, the given operation a ∗ b = a + ab is not commutative.

For a, b, c, we have

(a * b) * c = (a + ab) * c = a + ab + (a + ab)c = a + ab + ac + abc

a * (b * c) = a * (b + bc) = a + a(b + bc) = a + ab + abc

Since a + ab + ac + abc ≠ a + ab + abc, then (a * b) * c ≠ a * (b * c).

Thus, the given operation a ∗ b = a + ab is not associative.

(iv) a * b = (a – b)2

NCERT Solutions:

Given that ∗ be a binary operation as a ∗ b = (a – b)2

Since (a – b)2 = (b – a)2, then a * b = b * a

Thus, the given operation a ∗ b = (a – b)2 is commutative.

For a, b, c, we have

(a * b) * c = (a – b)2 * c = ((a – b)2 – c)2

a * (b * c) = a * (b – c)2 = (a – (b – c)2)2

Since ((a – b)2 – c)2 ≠ (a – (b – c)2)2, then (a * b) * c ≠ a * (b * c).

Thus, the given operation a ∗ b = (a – b)2 is not associative.

(v) a * b = ab/4

NCERT Solutions:

Given that ∗ be a binary operation as a ∗ b = ab/4

Since ab/4 = ba/4, then a * b = b * a

Thus, the given operation a ∗ b = ab/4 is commutative.

For a, b, c, we have Since abc/16 = abc/16, then (a * b) * c = a * (b * c). Thus, the given operation a ∗ b = ab/4 is associative.

(vi) a * b = ab2

NCERT Solutions:

Given that ∗ be a binary operation as a ∗ b = ab2

Since ab2 ≠ ba2, then a * b ≠ b * a

Thus, the given operation a ∗ b = ab2 is not commutative.

For a, b, c, we have

(a * b) * c = (ab2) * c = ab2c2

a * (b * c) = a * (bc2) = a(bc2)2 = ab2c4

Since ab2c2 ≠ ab2c4, then (a * b) * c ≠ a * (b * c).

Thus, the given operation a ∗ b = ab2 is not associative.

NCERT Solutions for Class 12 Maths Relations and Functions Exercise 1.4: Ques No 10. Show that none of the operations given above has identity.

NCERT Solutions:

We know that if a * e = a = e * a for all a ∈ Q, then the element e ∈ Q will be the identity element for the operation *.

However, there is no such element e ∈ Q with respect to each of the six operations given in question no 9.

Thus, none of the six operations in question 9 has identity.

NCERT Solutions for Class 12 Maths Relations and Functions Exercise 1.4: Ques No 11. Let A = N × N and ∗ be the binary operation on A defined by (a, b) ∗ (c, d) = (a + c, b + d). Show that ∗ is commutative and associative. Find the identity element for ∗ on A, if any.

NCERT Solutions:

Given that A = N × N and ∗ be the binary operation on A defined by

(a, b) ∗ (c, d) = (a + c, b + d).

For a, b, c, d ∈ N, we have

(a, b) ∗ (c, d) = (a + c, b + d) and (c, d) * (a, b) = (c + a, d + b) = (a + c, b + d)

Since (a + c, b + d) = (a + c, b + d), then (a, b) ∗ (c, d) = (c, d) * (a, b)

Thus, the given operation (a, b) ∗ (c, d) = (a + c, b + d) is commutative.

For a, b, c, d, e, f ∈ N, we have

{(a, b) ∗ (c, d)} * (e, f) = {a + c, b + d} * (e, f) = (a + c + e, b + d + f)

And,

(a, b) ∗ {(c, d) * (e, f)} = (a, b) * {c + e, d + f}= (a + c + e, b + d + f)

Thus, {(a, b) ∗ (c, d)} * (e, f) = (a, b) ∗ {(c, d) * (e, f)}.

The given operation (a, b) ∗ (c, d) = (a + c, b + d) is associative.

NCERT Solutions for Class 12 Maths Relations and Functions Exercise 1.4: Ques No 12. State whether the following statements are true or false. Justify.

(i) For an arbitrary binary operation ∗ on a set N, a ∗ a = a ∀ a ∈ N.

(ii) If ∗ is a commutative binary operation on N, then a ∗ (b ∗ c) = (c ∗ b) ∗ a

NCERT Solutions:

(i)

First we define an operation * on N as a * b = a + b ∀ a, b ∈ N.

Then, in particular, for b = a = 3, we have 3 * 3 = 3 + 3 = 6 ≠ 3

Therefore, statement (i) is false

(ii)

R.H.S.

= (c * b) * a

= (b * c) * a [* is commutative]

= a * (b * c) [Again, as * is commutative]

= L.H.S.

Therefore, statement (ii) is true.

NCERT Solutions for Class 12 Maths Relations and Functions Exercise 1.4: Ques No 13. Consider a binary operation ∗ on N defined as a ∗ b = a3 + b3. 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?

NCERT Solutions:

Given that ∗ be a binary operation as a ∗ b = a3 + b3

Since a3 + b3 = b3 + a3, then a * b = b * a

Thus, the given operation a ∗ b = a3 + b3 is commutative.

For a, b, c, we have

(a * b) * c = (a3 + b3) * c = (a3 + b3)3 + c3

a * (b * c) = a * (b3 + c3) = a3 + (b3 + c3)3

Since (a3 + b3)3 + c3 ≠ a3 + (b3 + c3)3, then (a * b) * c ≠ a * (b * c).

Thus, the given operation a ∗ b = a3 + b3 is not associative.

Therefore, the correct option is B.