B have a common transversal if and only if, for all , Generalizations A partial transversal is a set containing at most one element from each member of the collection, or (in the stricter form of the concept) a set with an injection from the set to C. The transversals of a finite collection C of finite sets form the basis sets of a matroid, the transversal matroid of C. The independent sets of the transversal matroid are the partial transversals of C. An independent transversal (also called a rainbow-independent set or independent system of representatives) is a transversal which is also an independent set of a given graph. To explain the difference in figurative terms, consider a faculty with m departments, where the faculty dean wants to construct a committee of m members, one member per department. Such a committee is a transversal. But now, suppose that some faculty members dislike each other and do not agree to sit in the committee together. In this case, the committee must be an independent transversal, where the underlying graph describes the "dislike" relations. Another generalization of the concept of a transversal would be a set that just has a non-empty intersection with each member of C. An example of the latter would be a Bernstein set, which is defined as a set that has a non-empty intersection with each set of C, but contains no set of C, where C is the collection of all perfect sets of a topological Polish space. As another example, let C consist of all the lines of a projective plane, then a blocking set in this plane is a set of points which intersects each line but contains no line. Category theory In the language of category theory, a transversal of a collection of mutually disjoint sets is a section of the quotient map induced by the collection. Computational complexity The computational complexity of computing all transversals of an input family of sets has been studied, in particular in the framework of enumeration algorithms. See also Axiom of choice Permanent References Further reading Lawler, E. L. Combinatorial Optimization: Networks and Matroids. 1976. Mirsky, Leon (1971). Transversal Theory: An account of some aspects of combinatorial mathematics. Academic Press. . Combinatorics Group theory Families of sets