Noncover

Noncover is a term used in mathematics and related disciplines to describe a structure that does not cover a given universal set or space. In set theory and combinatorics, let U be a universal set and F a collection (family) of subsets of U. F is a cover of U if the union of all members of F equals U. A noncover, then, is any situation in which the chosen collection fails to cover U; equivalently, the union of the subsets in the collection is a proper subset of U. In practical contexts, noncovers describe incomplete coverage in problems such as the set cover problem, where the objective is to select a subfamily whose union is U; any subfamily whose union is not U is a noncover of U.

In topology, a family of open sets that does not cover a space X is similarly described as a noncover of X by that family. In sensor networks and related algorithms, noncover arguments may be used to prove lower bounds on coverage or to identify uncovered regions.

Examples: U = {a, b, c, d}; F = { {a, b}, {c} } has union {a, b, c}, which is a noncover of U.

Etymology: from cover, with the prefix non- indicating negation.

See also: cover, set cover problem, covering, noncovering. Note that the term is not always standardized; in some texts, it is described procedurally as a non-covering or as a failure to achieve coverage by a given collection.