KMO 2017 Problem 1

Denote $U$ as the set of $20$ diagonals of the regular polygon $P_1P_2P_3P_4P_5P_6P_7P_8$ . Find the number of sets $S$ which satisfies the following conditions.

1. $S$ is a subset of $U$ .

2. If $P_iP_j \in S$ and $P_j P_k \in S$ , and $i \neq k$ , $P_iP_k \in S$ .