Question

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

On R − {−1}, define `a*b = a/(b+1)`

Answer 1

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.