For each opertion `**` difined below, determine whether `**` isw binary, commutative or associative.
(i) On Z, define `a **b =a - b`
(ii) On Q, define `a **b =ab +1`
(iii) On Q, define `a **b = (ab)/( 2)`
(iv) On `Z ^(+),` define `a **b = 2 ^(ab)`
(v) On `Z ^(+),` define `a **b=a ^(b)`
(vi) On `R - {-1}, `define `a **b= (a)/( b +1)`

Answer

(i) `**` is binary but neither commutative nor associative
(ii) `**` is binary, commutative but not associative
(iii) `**` is binary, both commutative and associative
(iv) `**` is binary, cimmutative but not associative
(v)` **` is binary but neither commutative nor associative
(vi) `**` not binary