The operation  ~$\bigotimes $~ is defined for all nonzero numbers a and b by ~$a\bigotimes b=\frac{a}{b}-\frac{b}{a}$~ .  If x and y are nonzero numbers, which of the following statements must be true?

I.  ~$x\bigotimes xy = x(1\bigotimes y)$~
II.  ~$x\bigotimes y = -(y\bigotimes x)$~
III.  ~$\frac{1}{x}\bigotimes \frac{1}{y} = y\bigotimes x$~