Transversal (combinatorics): Difference between revisions

Revision as of 12:52, 28 November 2005

Revision as of 23:21, 27 January 2006

Line 1:

{{otheruses}}

Given a collection ''C'' of disjoint [[set theory|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 firstthis (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''.

[[Image:Parallel_lines.png|thumb|right|250px|A transversal cutting two parallel lines]]

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 ==

Line 17 ⟶ 14:

[[Category:Combinatorics]]

[[Category:Group theory]]

[[Category:Set theory]]