Let $f(x_1,...,x_n)$ be a complex polynomial. Show the following two conditions on $f$ are equivalent: i) for any transpositions $\tau$ we have $\tau.f=-f$ and ii) for any $\sigma \in S_n$ we have $\sigma f=f$ when $\sigma$ is even and $\sigma .f=-f$ when $\sigma$ is odd. Such a polynomial is said to be anti-symmetric. Show that $\Delta (x_1,...x_n)$ is anti symmetric, where $\Delta (x_1,...x_n)$=$\Pi_{i<j}(x_i-x_j)$.
I should also note that $\tau.f=f(x_{\tau(1)},...x_{\tau(n)})$. I'm unsure how to define the function so I can start the proof. Furthermore for the last part the difference function is given as $(x_n-x_{n-1})(x_n-x_{n-2})....(x_2-x_1)$. I'm trying to find a relationship between the number of times a "swap" will occur upon a transposition and furthermore conclude that it's antisymmetric. I understand why $\Delta (x_1,...x_n)^2$ is symmetric though. Thank you in advanced.