Given a collection C of disjoint sets, a transversal is a set containing exactly one member of each of them. In case that the original sets are not disjoint, there are several variations. One variation is that there is a bijection f from the transversal to C such that x is an element of f(x) for each x in the transversal. Another is merely that the transversal must have non-empty intersection with each set in C.
As an example of the first (disjoint-sets) meaning of transversal, in group theory, given a subgroup H of a group G, a right (respectively left) transversal is a set containing exactly one element from each right (respectively left) coset of H.
A transversal in a Latin square of order n is a collection of n matrix positions comprising one from each row and one from each column, such that the symbols in those positions are distinct. Not all Latin squares have transversals.
In geometry, a transversal is a line that intersects two or more parallel lines. Such a line produces several congruent and several supplementary angles.
Reference
- Mirsky, Leon (1971). Transversal Theory. Academic Press. ISBN 0124985505.