Question

Exponential operation: (x, y) →xy on Set of Natural numbers and set of Integers

Answer 1

Consider the exponential operation $(x,y)\to x^y.$ If $x\in\N$ and $y\in \N$ then $x^y\in \N,$ and hence this operation is defined on the set $\N$ of natural numbers. On the other hand, for $x=2\in\Z$ and $y=-1\in\Z,$ we have that $x^y=2^{-1}=0.5\notin \N,$ and thus this operation is not defined on the set $\Z$ of integers.