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