The Average Size of Nonsingular Sets in a Graph

A nonsingular set in a finite graph is defined as the vertex set of an induced subgraph which has no isolated point. If G is a graph without isolated points and with at least two vertices and B is a connected subgraph, then the average size of those nonsingular sets in G which contain B is at least (|G| + |B|)/2. This result is used to prove the following: if F is a family of sets which is closed with respect to union, and if none of the generating sets in F has more than two elements, then the average size of a set in F is at least half of the size of the largest set in F.