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

Binary Operations Exercise Ex. 3.1

Solution 1(i)

Solution 1(ii)

Solution 1(iii)


Solution 1(iv)

Solution 1(v)

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 1(vii)

Solution 2

Solution 3

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

now,

3*4 = 2 × 3 + 4 - 3

      = 10 - 3

      = 7

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

Binary Operations Exercise Ex. 3.2

Solution 1


Solution 2

Solution 3

Solution 4(i)

Solution 4(ii)

Solution 4(iii)

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(v)

Solution 4(vi)

Solution 4(vii)


Solution 4(viii)

Solution 4(ix)

Solution 4(x)

Solution 4(xi)

Solution 4(xii)

Solution 4(xiii)

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 5

Solution 6

Solution 7

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)

Solution 9

Solution 9

Solution 10

Solution 11

Solution 12

Solution 13

Solution 14

Binary Operations Exercise Ex. 3.3

Solution 1

Solution 2

Solution 3

Solution 4

Binary Operations Exercise Ex. 3.4

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5


Solution 6

Solution 7

Solution 8

Binary Operations Exercise Ex. 3.5

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9


Solution 10

Binary Operations Exercise MCQ

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

 

Solution 7

Solution 8

 

Solution 9

Solution 10

Solution 11

Solution 12

Solution 13

Solution 14

Solution 15

Solution 16

Solution 17

Solution 18

Solution 19

Solution 20

Solution 21

Solution 22

Solution 23

Solution 24

Solution 25

 

Solution 26

Solution 27

Solution 28

Solution 29

Binary Operations Exercise Ex.3VSAQ

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

Solution 11

Solution 12

Solution 13

Solution 14

Solution 15

Solution 16

Solution 17

Solution 18

Solution 19

Solution 20

G i v e n space b i n a r y space o p e r a t i o n space i s space a asterisk times b equals 2 a plus b minus 3
T h u s comma space 3 asterisk times 4 equals 2 cross times 3 plus 4 minus 3 equals 6 plus 4 minus 3 equals 7


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