Bicontaining

Bicontaining is a term used in some discussions to describe mutual containment between two objects A and B: A contains B and B contains A. In standard set theory, this mutual containment implies A = B, since A ⊇ B and B ⊇ A together yield equality. Consequently, bicontaining is not a distinct relational concept for sets beyond the trivial case of equality; it can be seen as a way of expressing equality through containment in two directions.

Context and usage: The term is not widely standardized. When used, it is typically informal or illustrative rather than a formal axiom or definition. It may appear in expository writing to emphasize that two objects cannot differ without violating containment in at least one direction.

In other structures: If A and B are objects with a notion of containment (such as topological spaces with subspaces, or data structures with embedded components), bicontainment would correspond to mutual embedding that yields an isomorphism between A and B, effectively reducing to a statement of equivalence rather than a new relation.

Examples: A = B yields bicontainment; if A = B = the empty set, both directions hold. In geometric terms, two identical intervals on the real line are bicontaining with respect to the subset relation.

Limitations: As a formal notion, bicontaining is not widely adopted in standard literature and is best understood as a descriptive way to articulate mutual containment leading to equality.

See also: Containment, Inclusion, Subset, Equality, Isomorphism.

References: No canonical formal definition exists in standard references; usage is largely informal.