# Dictionary Definition

transversal adj : extending or lying across; in a crosswise direction; at right angles to the long axis; "cross members should be all steel"; "from the transverse hall the stairway ascends gracefully"; "transversal vibrations"; "transverse colon" [syn: cross(a), transverse, thwartwise]

# User Contributed Dictionary

## English

### Etymology

From transversalis.

1. Running or lying across; transverse; as, a transversal line.

### Noun

1. A straight line which traverses or intersects any system of other lines, as a line intersecting the three sides of a triangle or the sides produced.
2. A set containing one member from each of a collection of disjoint sets.

## French

### Etymology

From transversalis.

# Extensive Definition

In combinatorial mathematics, given a collection C of sets, a transversal is a set containing exactly one element from each member of the collection: it is a section of the quotient map induced by the collection. If the original sets are not disjoint, there are several different definitions. 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. A less restrictive definition requires that the transversal just has a non-empty intersection with each member of C.

## Examples

As an example of the 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.
Given a direct product of groups G = H \times K, then H is a transversal for the cosets of K, and conversely.
• The marriage theorem gives necessary and sufficient conditions for possibly overlapping subsets to have a transversal.

## References

• Mirsky, Leon (1971). Transversal Theory: An account of some aspects of combinatorial mathematics. Academic Press. ISBN 0-12-498550-5.
transversal in Spanish: Transversal (matemática)