Suppose [V.sub.1],..., [V.sub.m-1] are pairwise

disjoint sets in L = V U.

Among their problems are the Erdos-Ko-Rado theorem via shifting, the Kleitman theorem for no s pairwise

disjoint sets, uniform measure versus product measure, F'rude's structure theorem, some algebraic constructions for L systems, and a cross intersection problem with measures.

A bitopological space (X, [[tau].sub.1], [[tau].sub.2]) is called [[tau].sub.1][[tau].sub.2]-[delta] semiconnected space, if X cannot be expressed as the union of two

disjoint sets A([not equal to] [phi]) and B([not equal to] [phi]) such that (A [intersection] [[tau].sub.1]-[delta]scl(B)) [union] (B [intersection] [[tau].sub.2]-[delta]scl(A)) = [phi].

This problem differs from the target coverage which is based on disjoint covering sets, since coverage ensures that the [E.sub.relay] energy of each sensor from each of the

disjoint sets is kept at minimum.

(2) for every R > 0 and R' [greater than or equal to] 0 exists [p.sub.2] = [p.sub.2](R, R') [greater than or equal to] 1 that for all [z.sup.0] [member of] [C.sup.n] such that [T.sup.n]([z.sup.0], R/L([z.sup.0])) \ [G.sup.R'](F) = [[universal].sub.i] [C.sub.i] [not equal to] 0, where the sets [C.sub.i] are connected

disjoint sets, and either (a) [mathematical expression not reproducible], or (b) [mathematical expression not reproducible], or (c) [mathematical expression not reproducible], and [z.sup.*], [z.sup.**] belong to the same set [mathematical expression not reproducible]

The sensors can be divided into a collection of

disjoint sets such that every set can satisfy the coverage requirement.

Remark 3.2 If A and B are [alpha]-[tau]-separate sets, then both of them are also

disjoint sets.

A graph is an ordered pair of

disjoint sets such that is a subset of the set of unordered pairs of ; the set is the set of vertices and is the set of edges.

Near sets are

disjoint sets that resemble each other, especially resemblance defined within perceptual representative spaces (a.k.a., tolerance spaces).

Definition 4 A species (with restrictions) P is a Hopf monoid if there is a collection of maps [[??].sub.S,T]: [P.sub.S] x [P.sub.T] [right arrow] [P.sub.S[??]T] for any

disjoint sets S and T, which is natural in S and T, and is associative.