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Question
For each binary operation * defined below, determine whether * is commutative or associative.
On Z+, define a * b = 2ab
Solution
On Z+, * is defined by a * b = 2ab.
It is known that:
ab = ba &mnForE; a, b ∈ Z+
⇒ 2ab = 2ba &mnForE; a, b ∈ Z+
⇒ a * b = b * a &mnForE; a, b ∈ Z+
Therefore, the operation * is commutative.
It can be observed that:
`(1*2)*3 = 2^(1xx2) * 3 = 4 * 3 = 2^(4xx3) = 2^12`
`1 * (2 * 3) = 1 * 2^(2 xx 3) = 1 * 2^6 = 1 ** 64 = 2^(64)`
∴(1 * 2) * 3 ≠ 1 * (2 * 3) ; where 1, 2, 3 ∈ Z+
Therefore, the operation * is not associative.
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