Determine whether * on `N` defined by `a*b=a^b` for all `a ,\ b in N` define a binary operation on the given set or not:

Determine whether * on N defined by a*b=a+b-2 for all a ,\ b in N define a binary operation on the given set or not:

'*' on Q defined by a*b=(a-1)/(b+1) for all a ,\ b in Q define a binary operation on the given set or not:

Determine whether O on Z defined by a\ O\ b=a^b for all a ,\ b in Z define a binary operation on the given set or not:

Determine whether * on N defined by a*b=a+b-2 for all a,b in N define a binary operation on the given set or not:

^ prime O' on N defined by ab=a^(b)+b^(a) for all a,b in N define a binary operation on the given set or not:

:' on Q defined by a*b=(a-1)/(b+1) for all a,b in Q define a binary operation on the given set or not:

Determine whether O on Z defined by aOb=a^(b) for all a,b in Z define a binary operation on the given set or not:

'o' on N defined by a o b= a b + b a for all a, b∈N define a binary operation on the given set or not:

Determine whether '+_6' on S={0,\ 1,\ 2,\ 3,\ 4,\ 5} defined by a+_6b={a+b ,\ if\ a+b<6a+b-6,\ if\ a+bgeq6 define a binary operation on the given set or not:

Determine whether or not the definition of * On Z^+ , defined * by a*b=a b gives a binary operation. In the event that * is not a binary operation give justification of this. Here, Z^+ denotes the set of all non-negative integers.