Venn diagrams are useful devices for illustrating abstract concepts of algebraic operations defined on 'ordinary' sets. The basic oper- ations are union (if A and B are sets, this is the set of all elements in either A or B or both), intersection (the set of all elements in both A and B), complementation (the set of elements not contained in a set A, call A'), and inclusion (every element of A is an element of B). Note that this last operation can be interpreted as 'A implies B' or as 'B is implied by A'.
With respect to inclusion, clearly the city of Alberquerque is in the state of New Mexico. If two venn diagrams are drawn intersecting, then yes, this is logically equivalent to the inclusion statement, provided one of the nonintersecting circles has no elements, i.e., is empty.
The existence of the empty set follows logically as the set with no elements. Its size is zero. Equivalently, the probability of an impos- sible event is zero. Logically this is as fundamental as the concept of the number zero.
Speaking of sets and all that, what ever happened to Dolciani, et al.? Both my wife and I first learned of set theory from her series of textbooks (no, I didn't know my wife in HS).