Schröder-Bernstein theorem the theorem that mutually dominant sets are equinumerous. A set A is said to be dominated by a set B if and only if each element of A can be mapped to a unique element of B in such a way that no two elements of A are mapped to the same element of B (possibly with some elements of B left over). Intuitively, if A is dominated by B, then B has at least as many members as A. Given this intuition, one would expect that if A is dominated by B and B is dominated by A, then A and B are equinumerous (i.e., A can be mapped to B as described above with no elements of B left over). This is the Schröder-Bernstein theorem. Stated in terms of cardinal numbers, the theorem says that if k m l and l m k, then k % l. Despite the simplicity of the theorem’s statement, its proof is non-trivial. See also SET THEORY. P.Mad.