Determine whether or not each of the definition of * is binary, justify your answer.
(a) `a * b =a/(b+1)` on R-{-1}
(b) `a * b =a^(2) + b^(2) `on Q
( c) `a * b =(ab)/4` on Q
(d) `a * b =ab^(3)` on Q
(e) `a * b =(a-b)^(2)` on Q
( f) `a * b =2^(a-b)` on Z
( g) `a * b =ab + b -5` on Q

Answer

a) neither commutative, nor associative
b) commutative but not associative
c) commutative and associative
d) Neither commutative nor associative
e) Neither commutative nor associative
f) commutative but not associative
g) commutative but not associative
h) Neither commutative nor associative