Menu
Home
People
Places
Arts
History
Plants & Animals
Science
Life & Culture
Technology
Reference.org
Transversal (combinatorics)
open-in-new
<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, particularly in <a href="/facts/Combinatorics/k14xUoKT">combinatorics</a>, given a <a href="/facts/Family_of_sets/WbYof7lP">family of sets</a>, here called a collection <i>C</i>, a transversal (also called a cross-section) is a set containing exactly one element from each member of the collection. When the sets of the collection are mutually disjoint, each element of the transversal corresponds to exactly one member of <i>C</i> (the set it is a member of). If the original sets are not disjoint, there are two possibilities for the definition of a transversal: </p> <ul><li>One variation is that there is a <a href="/facts/Bijection/j4gTuTmW">bijection</a> <i>f</i> from the transversal to <i>C</i> such that <i>x</i> is an element of <i>f</i>(<i>x</i>) for each <i>x</i> in the transversal. In this case, the transversal is also called a system of distinct representatives (SDR).: 29 </li> <li>The other, less commonly used, does not require a one-to-one relation between the elements of the transversal and the sets of <i>C</i>. In this situation, the members of the system of representatives are not necessarily distinct.: 692 : 322 </li></ul> <p>In <a href="/facts/Computer_science/lN3pEyPz">computer science</a>, computing transversals is useful in several application domains, with the input <a href="/facts/Family_of_sets/WbYof7lP">family of sets</a> often being described as a <a href="/facts/Hypergraph/qIR2o2I3">hypergraph</a>. </p><p>In <a href="/facts/Set_theory/1ptfsvUP">set theory</a>, the <a href="/facts/Axiom_of_choice/1Ura2HTh">axiom of choice</a> is equivalent to the statement that every <a href="/facts/Partition_of_a_set/VeP9g8CA">partition</a> has a transversal. </p>