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 a⊙b = a2 + b2 for all a, b ϵ Q
Commutativity:
For a, b ϵ Q
a⊙b = a2 + b2 = b2 + a2 = b⊙a
So, '⊙' is commutative on Q.
Associativity:
For a, b, c ϵ Q
(a⊙b) ⊙c = (a2 + b2) ⊙c = (a2 + b2)2 + c2
a⊙(b ⊙c) =a ⊙( b2 +c2)= a2 +(b2 + c2 )2
(a⊙b) ⊙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
