For every $A \sim S$, let
$$
\beta_{\mathbf{A}}=\left\{\begin{array}{ll}
\sum_{\mathrm{a} \in \mathrm{A}}(-1)^{\mathrm{a}} a, & A \neq \varnothing, \\
0, & A=\varnothing .
\end{array}\right.
$$

Find $\sum_{\mathrm{A} \subset \mathrm{S}} \beta_{\mathrm{A}}$.
Take any $i \in S$, then the number of $A$ such that $i \in A \subset S$ is $2^{\mathrm{n}}-1$, so the contribution of $i$ to the expression $\sum_{\mathrm{A} \subset \mathrm{S}} \beta_{\mathrm{A}}$ is $(-1)^{1} \cdot i \cdot 2^{n-1}$. Therefore, we have
$$
\begin{array}{l}
\sum_{A \subset S} \beta_{1}=2^{n-1}\left(-1+2-3+\cdots+(-1)^{n} n\right) \\
=(-1)^{n}\left[\frac{n+1}{2}\right] 2^{D-1} .
\end{array}
$$