Let $T$ be a subset of $\{1,2,3,....n\}$ and let $A_T$ represent the number of ways of choosing a distinguished element from $T$. Let $B_T$ be the number of ways of choosing a distinguished element in $T$ and then a subset of the remaining elements. If counting a set in two different ways that $B_T=\sum_{S \subseteq T}A_S$. I was asked to figure out a formula for $A_T$ & $B_T$ for any subset of $T$ and then try to prove algebraically. I'm confused as to what $B_T$ is, much less the main point of the question.
How would one define the mobius function
For subsets $S$, $T \subseteq \{1,2,...,n\}$, define \begin{equation} \mu(S,T)= \begin{cases} (-1)^{|T|-|S|} & if S \subseteq T \\ 0 & otherwise \end{cases} \end{equation} and show that \begin{equation} \sum_{S\subseteq\{1,2,...n\}} \mu(S,T)= \begin{cases} 1 & if T = \{\} \\ 0 & otherwise \end{cases} \end{equation}
with the definition what exactly would be an example for $\mu(S,T)$