subsubset

Subsubset is not a standard term in formal set theory, but it is sometimes used informally to describe a subset that lies inside another subset. In most mathematical discourse, the ordinary terms subset and proper subset are preferred, and “subsubset” appears mainly in informal explanations, programming contexts, or educational writing to emphasize a hierarchical inclusion.

In a formal perspective, let S be a set and A a subset of S (A ⊆ S). A set B is described in some informal writings as a subsubset of S if B ⊆ A for some A with A ⊆ S. However, this condition is tautological: since B ⊆ A and A ⊆ S imply B ⊆ S, every subset B of S qualifies as a subsubset of S (by choosing A = B). Therefore, the phrase adds no extra mathematical structure beyond the ordinary notion that B is a subset of S. Some authors may attempt to restrict or specialize the idea (for example by requiring A to be nonempty or B to be a proper subset of A), but such refinements are not standardized.

Examples can illustrate the idea informally: take S = {1, 2, 3, 4} and A = {1, 2, 3}. Then B = {1, 2} is a subsubset of S via A, since B ⊆ A ⊆ S. Because A can always be chosen to contain B, any subset of S can be viewed as a subsubset in this loose sense.

Typical usage tends to favor precise language about inclusions and chains of subsets rather than the label subsubset.

See also: subset, proper subset, chain of subsets, power set, hierarchy.