Q. 55.0( 2 Votes )

If the binary ope

Answer :

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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