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Question
For each binary operation * defined below, determine whether * is commutative or associative.
On Z+, define a * b = ab
Solution
On Z+, * is defined by a * b = ab.
It can be observed that:
`1 *2 = 1^2 = 1` and `2 * 1 = 2^1 = 2`
∴ 1 * 2 ≠ 2 * 1 ; where 1, 2 ∈ Z+
Therefore, the operation * is not commutative.
It can also be observed that:
`(2 * 3)*4 = 2^3 * 4 = 8 *4 = 8^4 = (2^3)^4 = 2^(12)`
`2 * (3 *4) = 2 * 3^4 = 2 * 81 = 2^81`
∴(2 * 3) * 4 ≠ 2 * (3 * 4) ; where 2, 3, 4 ∈ Z+
Therefore, the operation * is not associative.
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