Does the operation * defined below on the given set represent a binary operation? : `a**b=(a+b)/2` on N

Does the operation * defined below on the given set represent a binary operation? : a**b=a+4b^2 on R

Does the operation * defined below on the given set represent a binary operation? : a**b=ab on Z^+

Does the operation * defined below on the given set represent a binary operation? : a**b=a-b on Z^+

Let ** on the set integers I be a binary operation defined by a**b=3a+4b-2 . Find 4**5 .

Let * be a binary operation defined by a*b = 3a+4b-2 find 4*5

Is the binary operation ** defined on the set of integer Z by the rule a**b=a-b+2 commutative ?

Is the binary operation ** defined on the set of rational numbers by the rule a**b=ab//4 associative ?

For the binary operation ** defined below, determine whether ** is commutative and associative : On N, define a**b=a^b

For the binary operation ** defined below, determine whether ** is commutative and associative : On N, define a**b=2^(ab)

For the binary operation ** defined below, determine whether ** is commutative and associative : On Q, define a**b=(ab)/2