transversaalsetele

Transversaalsetele is a term occasionally encountered in Dutch-language discussions of transversal concepts in set theory and combinatorics. It refers to the elements that make up a transversal across a given family of sets.

Definition: Suppose F = {F1, F2, ..., Fn} is a family of nonempty sets over a common universe U. A transversal is a subset T ⊆ U that intersects every Fi. The transversaalsetele are the elements that constitute a transversal, i.e., the members of T. There may be many transversaalsetele, corresponding to the different transversals of F. The smallest possible size of a transversal is called the transversal number τ(F).

Example: Let F = { {1,2}, {2,3}, {1,3} }. One transversal is T = {1,2}. Its transversaalsetele are 1 and 2. The transversal number τ(F) = 2.

Notes: In standard terminology, the usual name for the set that meets all Fi is simply a transversal, or a hitting set. The phrase transversaalsetele is not common in major references, and when used it usually means the elements of a transversal rather than a new object distinct from the transversal itself.

See also: transversal, hitting set, system of distinct representatives, SDR, hypergraph.

---