# 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,bN.

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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