Suppose that the sets are $A_1, A_2, \dots, A_{100}$. We claim that Mahdi needs $100$ steps. First of all, notice that $100$ would be enough; by calling $(1, 2)$, $(1, 3)$ and $(2, 3)$ he can understand $A_1$, $A_2$ and $A_3$. Because we have
$$A_1 = ((A_1 \cup A_2) - (A_2 \cup A_3)) \cup (A_1 \cap A_2) \cup (A_1 \cap A_3).$$
Then by calling $(i, 3)$ for $4 \le i \le 100$ he can determine all sets.

To prove that $100$ is necessary, consider a graph with $100$ vertices, and at each step connect vertices $i$ and $j$ if Mahdi calls $i$ and $j$. After $99$ steps the graph has $99$ edges so one of the connected components should be a tree. Hang the tree on a vertex and consider at each step $A_i \cup A_j = \{1\}$, $A_i \cap A_j = \emptyset$ then Mahdi can't distinguish between the case that all the sets at odd levels of the tree are empty and the sets on even levels are $\{1\}$ and the case where all the sets at odd levels are $\{1\}$ and the sets at even levels are empty. ■