Any permutation can be only either even or odd.

Question

A permutation can't be both even and Odd. How?? Is their any proof?? Kindly tell me.!

Thanks beforehand

Answer 1

Given a permutation $\pi$ we define $$\sigma(\pi) = |\{(i, j) : i < j \textrm{ and } \pi(i) > \pi(j)\}| \mod 2$$

It is easy to show inductively that $\sigma$ is $1\mod 2$ iff $\pi$ can be expressed as the product of a odd number of transpositions. By showing every transposition changes $\sigma$ by one. Once you have convinced yourself of this it follows that the parity of permutation is well defined.