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In combinatorial mathematics, given a collection ''C'' of disjoint 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''.

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Examples

Reference

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 $G\; =\; H\; imes\; K$, 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.

Reference

★ Mirsky, Leon (1971). ''Transversal Theory: An account of some aspects of combinatorial mathematics.'' Academic Press. ISBN 0-12-498550-5.