Question

Determine which of the following binary operations are associative and which are commutative: *on Q defined by a * b = \(\frac {a+b}{2}\) for all a,b Q.

Answer 1

Given that * is a binary operation on N defined by a * b = \(\frac {a+b}{2}\)  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:

∴ 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:

From (1) and (2) we can clearly say that associativity doesn’t hold for the binary operation ‘*’ on ‘N’.