I am confused on how to rigorously prove the solutions to this set of equations: $$ x^y= z, y^z= x, z^x = y $$ With that set of equations, the question states, prove that $x=y=z=1$ for x,y,z all in the positive reals?
What is the full solution to this problem?
If $x$, $y$, and $z$ are not all equal to $1$, then they either they are all greater than $1$ or are all less than $1$. In either case we have $x<z<y<x$—contradiction.