Consider an arbitrary directed graph G $.$ Let V be the set of all nodes of that graph and

F be a collection of sets F $$ V such that F $=/0,$ or F consists of a single node of G $,$ or

for each a$,$ b in F there is no directed path from a to b and no directed path from b to a$.$

Prove that $($V$,$ F $)$ is an independence system, but not a matroid.