# If the binary ope

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

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