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