I need some help with this problem, I really don't undertand how to start, thanks.
Let $S$ be a set. For each $i\in\mathbb{N}\cup\{0\}$ let $E_i$ the free module over $\mathbb{Z}$ generated by the elements $(x_0,...,x_i)\in S^{i+1}$. Define homomorphisms $d_{i+1}:E_{i+1}\rightarrow E_i$ as follows
$$d_{i+1}(x_0,...,x_{i+1})=\sum\limits_{j=0}^{i+1}(-1)^j(x_0,...,\hat{x}_j,...,x_i)$$
where $\hat{x}_j$ means that the term is omited. And for $i=0$ we define $d_{0}:E_{0}\rightarrow \mathbb{Z}$ as $d_0(x_0)=1$.
Show that it defines a complex
$$\rightarrow E_{i+1}\rightarrow E_i\rightarrow\dots\rightarrow E_0\rightarrow \mathbb{Z}\rightarrow0$$
where $E_0\rightarrow \mathbb{Z}\rightarrow0$ is exact.