Question

For each binary operation * defined below, determine whether * is commutative or associative.

On Q, define a * b  = `(ab)/2`

Answer 1

`On Q, * is defined by a * b  = `(ab)/2`

It is known that:

ab = ba &mnForE; a, b ∈ Q

⇒`"ab"/2 = "ba"/2` &mnForE; a, b ∈ Q

⇒ a * b = b * a &mnForE; a, b ∈ Q

Therefore, the operation * is commutative.

For all a, b, c ∈ Q, we have:

`(a*b)*c = ("ab"/2) * c = (("ab"/2)c)/2 = (abc)/4`

`a * (b*c) = a*("bc"/2) = (a("bc"/2))/2 = "abc"/4`

`∴(a * b) * c = a * (b * c)`

Therefore, the operation * is associative.