RELATIONS AND FUNCTIONS — Maths STD 12 Science — Question

We have,
a * b = a + b + ab for all a, b ∈ Q - {-1}
Commutativity: Let a, b ∈ Q - {-1}
⇒ a * b = a + b + ab = b + a + ba = b * a
⇒ a * b = b * a
⇒ '*' is commutative on Q - {-1}
Associativity: Let a, b, c ∈ Q - {-1}, then
⇒ (a * b) * c = (a + b + ab) * c
= a + b + ab + c + ac + bc + abc .......(i)
and, a * (b * c) = a * (b + c + bc)
= a + b + c + bc + ab + ac + abc .....(ii)
From (i) and (ii)
(a * b) * c = a * (b * c)
⇒ * is associative on Q - {-1}