 # 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:3

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:1

Given that

a*b=2a+b+ab+1 3*5=2×3+5+3×5+1

=6+5+15+1

=27

• Q3 Marks:3 • 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 = a2 + b2
(iii)a * b = a + ab
Find which of the binary operation are commutative and which are associative.

Marks:5

(i) Here a * b = a - b
Now b *a = b - a, but a - b b - a a * b b * 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 * b b * 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:3

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
1 a There does not exist any identity for this operation