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# Class 12-science RD SHARMA Solutions Maths Chapter 3 - Binary Operations

## Binary Operations Exercise Ex. 3.1

### Solution 1(vi)

We have,

A b = ab + ba for all a, b, ϵ N

Let a ϵ N and b ϵ N

ab ϵ N and ba ϵ N

ab + ba ϵ N

a b ϵ N

Thus, the operation ‘’ defines a binary relation on N

### Solution 3

It is given that, a*b = 2a + b - 3

now,

3*4 = 2 × 3 + 4 - 3

= 10 - 3

= 7

## Binary Operations Exercise Ex. 3.2

### Solution 4(iv)

'' on Q defined by ab = a2 + b2 for all a, b ϵ Q

Commutativity:

For a, b ϵ Q

ab = a2 + b2 = b2 + a2 = ba

So, '' is commutative on Q.

Associativity:

For a, b, c ϵ Q

(ab) ⊙c = (a2 + b2) ⊙c = (a2 + b2)2 + c2

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

(ab) ⊙c ≠ a⊙(b ⊙c)

So, '' is not associative on Q.

### Solution 4(xiv)

'*' on Q defined by a*b = a + b - ab for all a, b ϵ Z

Commutativity:

For a, b ϵ Z

a*b = a + b - ab = b + a - ba = b*a

So, '*' is commutative on Z.

Associativity:

For a, b, c ϵ Z

(a*b) *c = (a + b - ab) *c

= a + b - ab + c + ac + bc - abc

a*(b*c )= a*( b + c - bc)

= a + b +c - bc + ab + ac + - abc

(a*b) *c ≠ a*(b*c )

So, '*' is not associative on Z.

### Solution 4(xv)

'*' on Q defined by a*b = gcd (a, b) for all a, b ϵ N

Commutativity:

For a, b ϵ Q

a*b = gcd (a, b) = gcd (b, a) = b*a

So, '*' is commutative on N.

Associativity:

For a, b, c ϵ N

(a*b) *c = (gcd (a, b)) *c

= gcd (a, b, c)

=a*( gcd (b, c))

=a*(b*c)

(a*b) *c = a*(b*c )

So, '*' is associative on N.

### Solution 8

Now consider (a * b) * c.

Thus, we have, (a * b) * c = (a + b + ab) * c

= a + b + ab + c +(a + b + ab)c

= a + b + ab + c + ac + bc + abc

= a + b + c + ab + ac + bc + abc   ---(i)