Determine which of the following binary operations on the set N are associative and which are commutative.
`(a) a*b = 1AA a , b in N `
`(b) a*b = ((a+b)/2 )AA a , b in N`

Answer

(a) `*` is both associative and commutative.


(b) `*` is not associative.

Explanation

(a) By definition `a*b = b*a = 1, AA a, b in N`.
Also,

`(a*b)*c = (1*c) = 1` and `a*(b*c) = a*(1) = 1, AA a, b, c in N`.


Hence, R is both associative and commutative.


(b) `a*b = ((a+b)/2) = ((b+a)/2)=b*a`, shows that `*` is commutative.


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