SureDenNCRT Solution Exercise 4
Question 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
(ii) On Z+, define * by a * b = ab
(iii)On R, define * by a * b = ab2
(iv)On Z+, define * by a * b = |a − b|
(v) On Z+, define * by a * b = a
Answer 1:
(i) On Z+, * is defined by a * b = a − b.
It is not a binary operation
as the image of (1, 2) under * is 1 * 2 = 1 − 2 = −1 ∉ Z+.
(ii) On Z+, * is defined by a * b = ab.
It is seen that for each a, b ∈ Z+, there is a unique element ab in Z+.
This means that * carries each pair (a, b) to a unique element a * b = ab in Z+.
Therefore, * is a binary operation.
(iii)On R, * is defined by a * b = ab2
It is seen that for each a, b ∈ R, there is a unique element ab2 in R.
This means that * carries each pair (a, b) to a unique element a * b = ab2 in R.
Therefore, * is a binary operation.
(iv) On Z+, * is defined by a * b = |a − b|.
It is seen that for each a, b ∈ Z+, there is a unique element |a − b| in Z+.
This means that * carries each pair (a, b) to a unique element a * b = |a − b| in Z+.
Therefore, * is a binary operation.
(v) On Z+, * is defined by a * b = a.
It is seen that for each a, b ∈ Z+, there is a unique element a in Z+.
This means that * carries each pair (a, b) to a unique element a * b = a in Z+.
Therefore, * is a binary operation.
Question 2:
For each binary operation * defined below, determine whether * is commutative
or associative.
(i) On Z, define a * b = a − b
(ii) On Q, define a * b = ab + 1
(iii) On Q, define a * b =ab/2
(iv) On Z+, define a * b = 2ab
(v) On Z+, define a * b = ab
(vi)On R − {−1}, define a ∗ b =(a/b+1)
Answer 2:
(i) On Z, * is defined by a * b = a − b.
It can be observed that 1 * 2 = 1 − 2 = −1 and 2 * 1 = 2 − 1 = 1.
∴1 * 2 ≠ 2 * 1, where 1, 2 ∈ Z
Hence, the operation * is not commutative.
Also, we have
(1 * 2) * 3 = (1 − 2) * 3 = −1 * 3 = −1 − 3 = −4
1 * (2 * 3) = 1 * (2 − 3) = 1 * −1 = 1 − (−1) = 2
∴ (1 * 2) * 3 ≠ 1 * (2 * 3), where 1, 2, 3 ∈ Z
Hence, the operation * is not associative.
(ii) On Q, * is defined by a * b = ab + 1.
It is known that: ab = ba for all a, b ∈ Q
⇒ ab + 1 = ba + 1 for all a, b ∈ Q
⇒ a * b = a * b for all a, b ∈ Q
Therefore, the operation * is commutative.
It can be observed that
(1 * 2) * 3 = (1 × 2 + 1) * 3 = 3 * 3 = 3 × 3 + 1 = 10
1 * (2 * 3) = 1 * (2 × 3 + 1) = 1 * 7 = 1 × 7 + 1 = 8
∴ (1 * 2) * 3 ≠ 1 * (2 * 3), where 1, 2, 3 ∈ Q
Therefore, the operation * is not associative.
(iii) On Q, * is defined by a * b =(ab/2)
It is known that: ab = ba for all a, b ∈ Q
⇒(ab/2)=(ba/2) for all a, b ∈ Q
⇒ a * b = b * a for all a, b ∈ Q
Therefore, the operation * is commutative.
For all a, b, c ∈ Q, we have
= (a ∗b)*c = (ab/2)*c = ((ab/2)c) / 2 = (abc / 4)
And
= a ∗(b*c) = a*(bc/2) = (a(bc/2)) / 2 = (abc / 4)
∴ (a*b)*c = a*(b*c), where a, b, c ∈ Q
Therefore, the operation * is associative.
(iv) On Z+, * is defined by a * b = 2ab.
It is known that: ab = ba for all a, b ∈ Z+
⇒ 2ab= 2ba for all a, b ∈ Z+
⇒ a * b = b * a for all a, b ∈ Z+
Therefore, the operation * is commutative.
It can be observed that
(1 ∗ 2) ∗ 3 = 21×2∗ 3 = 4 ∗ 3 = 24×3 = 212 and
1 ∗ (2 ∗ 3) = 1 ∗ 22×3 = 1 ∗ 26 = 1 ∗ 64 = 21×64 = 264
∴ (1 * 2) * 3 ≠ 1 * (2 * 3), where 1, 2, 3 ∈ Z+
Therefore, the operation * is not associative.
(v) On Z+, * is defined by a * b = ab
It can be observed that
1*2 = 12 = 1 and 2*1 = 21 = 2
∴ 1 * 2 ≠ 2 * 1, where 1, 2 ∈ Z+
Therefore, the operation * is not commutative.
It can also be observed that
(2 ∗ 3) ∗ 4 = 23 ∗ 4 = 8 ∗ 4 = 84 = 212 and
2 ∗ (3 ∗ 4) = 2 ∗ 34 = 2 ∗ 81 = 281
∴ (2 * 3) * 4 ≠ 2 * (3 * 4), where 2, 3, 4 ∈ Z+
Therefore, the operation * is not associative.
(vi) On R, * − {−1} is defined by a ∗ b = (a / b+1)
It can be observed that
1 ∗ 2 =(1 /(2+1))=1/3 and 2 ∗ 1 =(2 / (1+1))=(2/2)= 1
∴1 * 2 ≠ 2 * 1, where 1, 2 ∈ R − {−1}
Therefore, the operation * is not commutative.
It can also be observed that
(1 ∗ 2) ∗ 3 =(1 / (2 + 1))∗ 3 =(1/3)∗ 3 =(1/3) / (3 + 1)=(1/12)
and
1 ∗ (2 ∗ 3) = 1 ∗(2 / (3 + 1))= 1 ∗(2 / 4)= 1 ∗(1/2) = (1/((1/2)+ 1)=((1/3) /2)=2/3
∴ (1 * 2) * 3 ≠ 1 * (2 * 3), where 1, 2, 3 ∈ R − {−1}
Therefore, the operation * is not associative.
Question 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 ∧.
Answer 3:
The binary operation ∧ on the set {1, 2, 3, 4, 5} is defined as a ∧ b = min {a, b}
for all a, b ∈ {1, 2, 3, 4, 5}.
Thus, the operation table for the given operation ∧ can be given as:
|
∧ |
1 |
2 |
3 |
4 |
5 |
|
1 |
1 |
1 |
1 |
1 |
1 |
|
2 |
1 |
2 |
2 |
2 |
2 |
|
3 |
1 |
3 |
3 |
3 |
3 |
|
4 |
1 |
2 |
3 |
4 |
4 |
|
5 |
1 |
2 |
3 |
4 |
5 |
Question 4: Consider a binary operation * on the set {1, 2, 3, 4, 5} given by the following
multiplication table.
(i) Compute (2 * 3) * 4 and 2 * (3 * 4)
(ii) Is * commutative?
(iii) Compute (2 * 3) * (4 * 5).
(Hint: use the following table)
|
* |
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 |
Answer 4:
(i) (2 * 3) * 4 = 1 * 4 = 1
2 * (3 * 4) = 2 * 1 = 1
(ii) For every a, b ∈ {1, 2, 3, 4, 5}, we have a * b = b * a. Therefore, the operation * is commutative.
(iii) (2 * 3) = 1 and (4 * 5) = 1
∴ (2 * 3) * (4 * 5) = 1 * 1 = 1
Question 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.
Answer 5:
The binary operation *′ on the set {1, 2, 3 4, 5} is defined as a *′ b = H.C.F of a and b.
The operation table for the operation *′ can be given as:
|
*’ |
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 tables for the operations * and *′ are the same.
Thus, the operation *′ is same as the operation*.
Question 6:
Let * be the binary operation on N given by a * b = L.C.M. of a and b. 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 *?
Answer 6:
The binary operation * on N is defined as a * b = L.C.M. of a and b.
(i) 5 * 7 = L.C.M. of 5 and 7 = 35
20 * 16 = L.C.M of 20 and 16 = 80
(ii) It is known that
L.C.M of a and b = L.C.M of b and a for all a, b ∈ N.
∴ a * b = b * a
Thus, the operation * is commutative.
(iii) For a, b, c ∈ N, 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
∴ (a * b) * c = a * (b * c)
Thus, the operation * is associative.
(iv)It is known that:
L.C.M. of a and 1 = a = L.C.M. 1 and a for all a ∈ N
⇒ a * 1 = a = 1 * a for all a ∈ N
Thus, 1 is the identity of * in N.
(v) An element a in N is invertible with respect to the operation * if there exists
an element b in N, such that a * b = e = b * a.
Here, e = 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 *.
Question 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.
Answer 7:
The operation * on the set A = {1, 2, 3, 4, 5} is defined as a * b = L.C.M. of a and b.
Then, the operation table for the given operation * can be given as:
|
* |
1 |
2 |
3 |
4 |
5 |
|
1 |
1 |
2 |
3 |
4 |
5 |
|
2 |
2 |
2 |
6 |
4 |
10 |
|
3 |
3 |
6 |
3 |
12 |
15 |
|
4 |
4 |
4 |
12 |
4 |
20 |
|
5 |
5 |
10 |
15 |
20 |
5 |
It can be observed from the obtained table that
3 * 2 = 2 * 3 = 6 ∉ A,
5 * 2 = 2 * 5 = 10 ∉ A,
3 * 4 = 4 * 3 = 12 ∉ A,
3 * 5 = 5 * 3 = 15 ∉ A,
4 * 5 = 5 * 4 = 20 ∉ A
Hence, the given operation * is not a binary operation.
Question 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?
Answer 8:
The binary operation * on N is defined as: a * b = H.C.F. of a and b
It is known that
H.C.F. of a and b = H.C.F. of b and a for all a, b ∈ N.
∴ a * b = b * a
Thus, the 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
∴ (a * b) * c = a * (b * c)
Thus, the 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 operation * does not have any identity in N.
Question 9: Let * be a binary operation on the set Q of rational numbers as follows:
(i) a * b = a − b
(ii) a * b = a2+ b2
(iii) a * b = a + ab
(iv) a * b = (a − b)2
(v) a ∗ b =(ab/4)
(vi) a * b = ab2
Find which of the binary operations are commutative and which are associative.
Answer 9: (i) On Q, the operation * is defined as a * b = a − b. It can be observed that:
(1/2)∗(1/3)=(1/2)−(1/3) = ((3 – 2)/6) = (1/6)
and
(1/3)∗(1/2)=(1/3)−(1/2)=((2 – 3)/6)=(−1/6)
∴(1/2)∗(1/3)≠(1/3)∗(1/2), where (1/2),(1/3)∈ Q
Thus, the operation * is not commutative.
It can also be observed that
((1/2)∗(1/3)) ∗ (1/4) = ((1/2)−(1/3)) ∗ (1/4) = ((3 – 2) / 6) ∗ (1/4) = (1/6) ∗(1/4) = ((1/6)−(1/4)) = ((2 – 3) /12) = (−1/12)
And
(1/2)∗((1/3) ∗ (1/4)) = (1/2) ∗ ((1/3) - (1/4)) = (1/2) ∗ ((4 – 3) / 12) = (1/2) ∗ (1/12) = (1/2) - (1/12) = ((6 – 1) /12) = (−1/12)
∴(1/2)∗ (1/3)∗ (1/4) ≠ (1/2) ∗((1/3)∗ (1/4)) , where (1/2), (1/3), (1/4) ∈ Q
Thus, the operation * is not associative.
(ii) On Q, the operation * is defined as a * b = a2+ b2.
For a, b ∈ Q, we have
a * b = a2+ b2= b2+ a2 = b * a
∴ a * b = b * a
Thus, the operation * is commutative.
It can be observed that
(1 * 2) * 3 = (12+ 22) * 3 = (1 + 4) * 3 = 5 * 3 = 52+ 32= 34 and
1 * (2 * 3) = 1 * (22+ 32) = 1 * (4 + 9) = 1 * 13 = 12+ 132=170
∴ (1 * 2) * 3 ≠ 1 * (2 * 3), where 1, 2, 3 ∈ Q
Thus, the operation * is not associative.
(iii)On Q, the operation * is defined as a * b = a + ab.
It can be observed that
1 * 2 = 1 + 1×2 = 1 + 2 = 3
2 * 1 = 2 + 2×1 = 2 + 2 = 4
∴ 1 * 2 ≠ 2 * 1, where 1, 2 ∈ Q
Thus, the operation * is not commutative.
It can also be observed that
(1 * 2) * 3 = (1+ 1×2 ) * 3 = (1 + 2) * 3 = 3 * 3 = 3 + 3×3 = 3 + 9 = 12 and
1 * (2 * 3) = 1 * (2 + 2×3 ) = 1 * (2 + 6) = 1 * 8 = 1 + 1×8 =1 + 8 = 9
∴ (1 * 2) * 3 ≠ 1 * (2 * 3), where 1, 2, 3 ∈ Q
Thus, the operation * is not associative.
(iv)On Q, the operation * is defined by a * b = (a − b)2.
For a, b ∈ Q, we have
a * b = (a − b)2
b * a = (b − a)2= [− (a − b)]2= (a − b)2
∴ a * b = b * a
Thus, the operation * is commutative.
It can be observed that
(1 * 2) * 3 = (1 – 2)2* 3 = (– 1)2* 3 = 1 * 3 = (1 – 3)2= (– 2)2= 4
And
1 * (2 * 3) = 1 * (2 – 3)2= 1 * (– 1)2= 1 * 1 = (1 – 1)2= 0
∴ (1 * 2) * 3 ≠ 1 * (2 * 3), where 1, 2, 3 ∈ Q
Thus, the operation * is not associative.
(v) On Q, the operation * is defined as a*b = (ab/4).
For a, b ∈ Q, we have
a*b = (ab/4) = (ba/4) = b*a
∴ a * b = b * a
Thus, the operation * is commutative.
For a, b, c ∈ Q, we have
(a*b) *c = (ab/4) *c = (((ab/4).c) / 4) = (abc / 16)
And
a*(b *c) = a*(bc/4) = ((a(bc/4)) / 4) = (abc / 16)
∴ (a * b) * c = a * (b * c), where a, b, c ∈ Q
Thus, the operation * is associative.
(vi) On Q, the operation * is defined as a * b = ab2
It can be observed that
(1/2)∗(1/3) = (1/2). (1/3)2 =(1/2).(1/9) = (1/18)
And
(1/3)∗(1/2) = (1/3). (1/2)2 =(1/3).(1/4) = (1/12)
∴(1/2)*(1/3) ≠ (1/3) * (1/2), where (1/2) and (1/3) ∈ Q
Thus, the operation * is not commutative.
It can also be observed that
((1/2)*(1/3)) * (1/4) =[(1/2)(1/3)2]∗ (1/4) = (1/18).(1/4)2 =(1 / (18 × 16)) = (1/288)
And
(1/2)*((1/3)* (1/4)) =(1/2)*[(1/3)∗ (1/4)2] = (1/2).(1/48) =(1/2) (1 × 48)2 = (1/(2 x 2304)) = (1 / 4608)
∴ (1/2)*(1/3) * (1/4) ≠ (1/2) * ((1/3)∗ (1/4)), where (1/2),(1/3), (1/4) ∈ Q
Thus, the operation * is not associative.
Hence, the operations defined in (ii), (iv), (v) are commutative and the operation defined in (v) is associative.
Question 10: Find which of the operations given above has identity.
Answer 10:
An element e ∈ Q will be the identity element for the operation *
if a * e = a = e * a, for all a ∈ Q.
However, there is no such element e ∈ Q with respect to each of the six operations satisfying the above condition.
Thus, none of the six operations has identity.
Question 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.
Answer 11:
A = N × N and * is a binary operation on A and is defined by
(a, b) * (c, d) = (a + c, b + d)
Let (a, b), (c, d) ∈ A
Then, a, b, c, d ∈ N
We have:
(a, b) * (c, d) = (a + c, b + d)
(c, d) * (a, b) = (c + a, d + b) = (a + c, b + d)
[Addition is commutative in the set of natural numbers]
∴ (a, b) * (c, d) = (c, d) * (a, b)
Therefore, the operation * is commutative.
Now, let (a, b), (c, d), (e, f) ∈ A
Then, a, b, c, d, e, f ∈ N
We have
[(a, b) * (c,d)] * (e, f) = (a + bc + 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)
∴[(a, b) * (c,d)] * (e, f) = (a, b) * [(c,d) * (e, f)]
Therefore, the operation * is associative.
Let an element e = (e1, e2) ∈ A will be an identity element for the operation *
if a * e = a = e * a for all a = (a1, a2) ∈ A
i.e., (a1 + e1, a2 + e2) = (a1, a2) = (e1 + a1, e2 + a2)
Which is not true for any element in A.
Therefore, the operation * does not have any identity element.
Question 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
Answer 12:
(i) 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.
∴ a * (b * c) = (c * b) * a
Therefore, statement (ii) is true.
Question 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?
Answer 13:
On N, the operation * is defined as a * b = a3+ b3.
For, a, b, ∈ N, we have
a * b = a3+ b3= b3+ a3
= b * a [Addition is commutative in N]
Therefore, the operation * is commutative.
It can be observed that
(1 * 2) * 3 = (13+ 23) * 3 = (1 + 8) * 3 = 9 * 3 = 93+ 33= 729 + 27 = 756 and
1* (2 * 3) = 1 * (23+ 33) = 1* (8 + 27) = 1*35 = 13+ 353= 1 + 42875 = 42876
∴ (1 * 2) * 3 ≠ 1 * (2 * 3), where 1, 2, 3 ∈ N
Therefore, the operation * is not associative.
Hence, the operation * is commutative, but not associative.
Thus, the correct answer is B.