Q. 124.3( 8 Votes )
Check the commutativity and associativity of each of the following binary operations:
‘*’ on N defined by a*b = 2ab for all a,b∈N
Given that * is a binary operation on N defined by a*b = 2ab for all a,b∈N.
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 = 2ab
⇒ b*a = 2ba = 2ab
⇒ b*a = a*b
∴ The commutative property holds for given binary operation ‘*’ on ‘N’.
We know that associative property is (p*q)*r = p*(q*r)
Let’s check the associativity of given binary operation:
⇒ (a*b)*c = (2ab)*c
⇒ ...... (1)
⇒ a*(b*c) = a*(2bc)
⇒ ...... (2)
From (1) and (2) we can clearly say that associativity doesn’t hold for the binary operation ‘*’ on ‘N’.
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