Check the commutativity and associativity of each of the following binary operations:
‘*’ on R defined by a*b = a + b – 7 for all a,b∈Q
Given that * is a binary operation on R defined by a*b = a + b – 7 for all a,b∈R.
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’.