Q. 4 N5.0( 1 Vote )

Check the commutativity and associativity of each of the following binary operations:

‘*’ on Z defined by a*b = a + b – ab for all a,bZ

Answer :

Given that * is a binary operation on Z defined by a*b = a + b – ab for all a,bZ.


We know that commutative property is p*q = q*p, where * is a binary operation.


Let’s check the commutativity of given binary operation:


a*b = a + b – ab


b*a = b + a – ba = a + b – ab


b*a = a*b


Commutative property holds for given binary operation * on Z.


We know that associative property is (p*q)*r = p*(q*r)


Let’s check the associativity of given binary operation:


(a*b)*c = (a + b – ab)*c


(a*b)*c = a + b – ab + c – ((a + b – ab)×c)


(a*b)*c = a + b + c – ab – ac – bc + abc ...... (1)


a*(b*c) = a*(b + c – bc)


a*(b*c) = a + b + c – bc – (a×(b + c – bc))


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


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


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