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

‘*’ on R defined by a*b = a + b – 7 for all a,bQ

Given that * is a binary operation on R defined by a*b = a + b – 7 for all a,bR.


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 – 7


b*a = b + a – 7 = a + b – 7


b*a = a*b


Commutative property holds for given binary operation * on R.


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 – 7)*c


(a*b)*c = a + b – 7 + c – 7


(a*b)*c = a + b + c – 14 ...... (1)


a*(b*c) = a*(b + c – 7)


a*(b*c) = a + b + c – 7 – 7


a*(b*c) = a + b + c – 14 ...... (2)


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


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