Given that ο is a binary operation on Q – { – 1} defined by aοb = a + b – ab for all a,b∈Q – { – 1}.

We know that commutative property is pοq = qοp, where ο is a binary operation.

Let’s check the commutativity of given binary operation:

⇒ aοb = a + b – ab

⇒ bοa = b + a – ba = a + b – ab

⇒ b*a = a*b

∴ Commutative property holds for given binary operation ‘ο’ on ‘Q – { – 1}’.

We know that associative property is (pοq)οr = pο(qοr)

Let’s check the associativity of given binary operation:

⇒ (aοb)οc = (a + b – ab)οc

⇒ (aοb)οc = a + b – ab + c – ((a + b – ab)×c)

⇒ (aοb)οc = a + b + c – ab – ac – ab + abc ...... (1)

⇒ aο(bοc) = aο(b + c – bc)

⇒ aο(bοc) = a + b + c – bc – (a×(b + c – bc))

⇒ a*(b*c) = a + b + c – ab – bc – ac + abc ...... (2)

From (1) and (2) we can clearly say that associativity hold for the binary operation ‘*’ on ‘Q – { – 1}’.