Binary Operations
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Q1
Consider the binary operation * on the set {1, 2, 3, 4, 5} given by the following multiplication table. Using this table compute
*
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
(i) (3*4)*5.
(ii) 3*(4*5).
(iii) (2*3)*(4*5).
(iv) if * is commutative.Marks:3Answer:
We have
(i) (3*4)*5 = 1*5 [Since, 3*4=1]=1
(ii) 3*(4*5) = 3*1 [Since, 4*5=1]=1
(iii) (2*3)*(4*5)
= 1*1 [ 2*3=1 and 4*5=1]
=1
(iv) Since(2*5) = 1 and
(5*2) =1
(2*5) = (5*2)
Operation * is commutative. 
Q2
Let * be a binary operation defined by a*b=2a+b+ab+1. Find 3*5.
Marks:1Answer:
Given that a*b=2a+b+ab+1 3*5=2 × 3+5+3 × 5+1 =6+5+15+1 =27 
Q3Marks:3
Answer:

Q4
Let * be the binary operation on the set Q of rational numbers which are as follows:
(i)a * b = a  b
(ii)a * b = a^{2} + b^{2}
(iii)a * b = a + ab
Find which of the binary operation are commutative and which are associative.Marks:5Answer:
(i) Here a * b = a  b
Now b *a = b  a, but a  bb  a
a * bb * a
* is not commutative
a*(b*c) = a*(b  c) = a  (b  c) = a  b + c
(a*b)*c = (a  b)*c = (a  b)  c
Thus, a*(b*c)(a * b)*c
* is not associative
(ii) Here a*b = a2 + b2
b*a = b2 + a2 = a2 + b2
a*b = b*a
* iscommutative
a*(b*c) = a*(b2+ c2) = a2+ (b2+ c2)2
(a*b)*c = (a2+ b2)*c = (a2 + b2)2 + c2
Thus, a*(b*c)(a*b)*c
* is not associative
(iii) Here a * b = a+ ab
Now b *a = b+ ba
a * bb * a
* is not commutative
a*(b*c) = a*(b+bc) = a+ a(b+ bc) = a+ ab + abc
(a*b)*c = (a+ ab)*c = a+ ab+ (a + ab)c = a + ab + ac + abc
Thus, a*(b*c)(a * b)*c
* is not associative 
Q5
Let the * binary operation on N be defined by a*b = HCF of a and b. Is * commutative? Is * associative? Does there exist identity for this operation on N?
Marks:3Answer:
Here a*b = HCF of a and b.
(a) We knowHCF of a, b = HCF of b, a
a*b = b*a is commutative
(b)a*(b*c) = a*(HCF of b, c)
= HCF of a and HCF of b, c
= HCF of a, b and c
Now(a*b)*c = (HCF of a, b)*c
= HCF of a, b and HCF of c
= HCF of a, b and c = a*(b*c) = (a*b)*c
* is associative(c)1*a = a*1 = HCF of a and 1
i.e., 1*a = a*1 = 1
1aThere does not exist any identity for this operation