Discuss the commutativity and associativity of binary operation '*' defined on A = Q − {1} by the rule a * b = a − b + ab for all, a, b ∊ A. Also find the identity element of * in A and hence find the invertible elements of A.
Given, * is a binary operation on Q − {1} defined by a*b=a−b+ab
Commutativity:
For any a, b∈A,
we have a*b=a−b+ab and b*a=b−a+ba
Since, a−b+ab≠b−a+ab
∴a*b≠b*a
So, * is not commutative on A.
Associativity:
Let a, b, c∈A(a*b)*c=(a−b+ab)*c
⇒(a*b)*c=(a−b+ab)−c+(a−b+ab)c
⇒(a*b)*c=a−b+ab−c+ac−bc+abc
a*(b*c)=a*(b−c+bc)
⇒a*(b*c)=a−(b−c+bc)+a(b−c+bc)
⇒a*(b*c)=a−b+c−bc+ab−ac+abc
⇒(a*b)*c≠a*(b*c)
So, * is not associative on A.
Identity Element
Let e be the identity element in A, then
a*e=a=e*a ∀a∈Q−{1}
⇒a−e+ae=a
⇒(a−1)e=0
⇒e=0 (As a≠1)
So, 0 is the identity element in A.
Inverse of an Element
Let a be an arbitrary element of A and b be the inverse of a. Then,
a*b=e=b*a
⇒a*b=e
⇒a−b+ab=0 [∵e=0]
⇒a=b(1−a)
⇒b=a/1−a
Since b∈Q−1
So, every element of A is invertible.
