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. Combinatorics is a branch of Pure mathematics concerning the study of discrete (and usually finite) objects In Category theory, a branch of Mathematics, a section is a right inverse of a morphism In Topology and related areas of Mathematics, a quotient space (also called an identification space) is intuitively speaking the result of identifying 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. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property 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. Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of In Mathematics, if G is a group, H is a Subgroup of G, and g is an element of G, then gH

Given a direct product of groups , then H is a transversal for the cosets of K, and conversely. In Mathematics, one can often define a direct product of objectsalready known giving a new one

• The marriage theorem gives necessary and sufficient conditions for possibly overlapping subsets to have a transversal. In Mathematics, the marriage theorem (1935 usually credited to mathematician Philip Hall, is a combinatorial result that gives the condition allowing

References

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

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