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Question
For each binary operation * defined below, determine whether * is commutative or associative.
On Q, define a * b = `(ab)/2`
Solution
`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.
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