# Inclusion exclusion similar problem

Inclusion-exclusion principle was formulated in my school in this way:

Let $A_i,\ldots,A_n \subseteq X$, where $X$ is a finite set. Let

$$S_r=\displaystyle \sum_{1 \le i_1< \cdots < i_r \le n} |A_{i_1}\cap \cdots \cap A_{i_r}|,\text{ and }S_0=|X|.$$

If $D(k)$ is number of elements belonging to exactly $k$ sets (from $A_i,\ldots,A_n$), then we have: $D(k)=\sum_{r\ge k}{r\choose k}(-1)^{r-k}S_r$.

Prove that if $\displaystyle A(r)=\sum_{j=r+1}^n (-1)^jS_j$ then $A(r)\le 0$ when $r$ is even and $A(r)\ge 0$ when $r$ is odd.