Question about permutation studying alternating group

Question

I'm studying alternating group with the textbook written by Dummit & Foote. For $\sigma \in S_n$ Ch3.5, defined $$\Delta = \prod_{1\leq i< j\leq n}(x_i-x_j)$$ $$\sigma(\Delta)=\prod_{1\leq i< j\leq n}(x_{\sigma(i)}-x_{\sigma(j)})$$ And they said 'Since $\sigma$ is a bijection of the indices, $\sigma(\Delta)$ must contains either $x_i-x_j$ or $x_j-x_i$'. I have trouble with understanding this sentence. I already know only that $\sigma$ is a bijection. Can you help me understanding this?

Answer 1

Let $i=1$, and $j=2$, wlog. Since $\sigma$ is a bijection, there are $k,l$ with $\sigma(k)=1$ and $\sigma (l)=2$. Now either $k\lt l$ or $l\lt k$ (since $k\ne l$). If $k\lt l$, then we have$x_{\sigma(k)}-x_{\sigma(l)}=x_1-x_2$. Otherwise we have $x_2-x_1$.