Proposition

In an FBAS with quorum intersection, the set of befouled nodes is a DSet.

Solution

Let $\ev{V, Q}$ be an FBAS with quorum intersection. Let $B$ be the intersection of all DSets that contain all ill-behaved nodes. Since the intersection of two DSets is a DSet and $V$ is finite, $B$ is a DSet.

• $v \in B$.
  • Then there exists no DSet $B_v$ such that $B_v$ contains all ill-behaved nodes and $v \notin B_v$. Therefore, $v$ is not an intact node. In other words, $v$ is a befouled node.
• $v \notin B$.
  • Then there exists a DSet $B_v$ that contains all ill-behaved nodes and $v \notin B_v$. In other words, $v$ is intact and thus $v$ is not a befouled node.

Therefore, $B$ is precisely the set of befouled nodes and it is a DSet.