Editing Hall's marriage theorem (section)

===Statement===
Let <math>\mathcal F</math> be a finite [[Family of sets|family]] of sets (note that although <math>\mathcal F</math> is not itself allowed to be infinite, the sets in it may be so, and <math>\mathcal F</math> may contain the same set [[multiplicity (mathematics)|multiple times]]).<ref>{{harvnb|Hall|1986|loc=pg. 51}}. An alternative form of the marriage theorem applies to finite families of sets that can be infinite. However, the situation of having an infinite number of sets while allowing infinite sets is not allowed.</ref> Let <math>X</math> be the union of all the sets in <math>\mathcal F</math>, the set of elements that belong to at least one of its sets. A '''[[Transversal (combinatorics)|transversal]]''' for <math>\mathcal F</math> is a subset of <math>X</math> that can be obtained by choosing a distinct element from each set in <math>\mathcal F</math>. This concept can be formalized by defining a transversal to be the [[Image (mathematics)|image]] of an [[injective function]] <math>f:\mathcal F\to X</math> such that <math>f(S)\in S</math> for each <math>S\in\mathcal F</math>. An alternative term for ''transversal'' is ''system of distinct representatives''.

The collection <math>\mathcal F</math> satisfies the '''marriage condition''' when each subfamily of <math>\mathcal F</math> contains at least as many distinct members as its number of sets. That is, for all <math>\mathcal G \subseteq \mathcal F</math>,
<math display=block>|\mathcal G|\le\Bigl|\bigcup_{S\in \mathcal G} S\Bigr|.</math>
If a transversal exists then the marriage condition must be true: the function <math>f</math> used to define the transversal maps <math>\mathcal G</math> to a subset of its union, of size equal to <math>|\mathcal G|</math>, so the whole union must be at least as large. Hall's theorem states that the converse is also true:

{{math_theorem
| name = Hall's Marriage Theorem
| math_statement = A family <math>\mathcal F</math> of finite sets has a transversal if and only if <math>\mathcal F</math> satisfies the marriage condition.
}}The name "marriage theorem" came from {{Harvard citation|Halmos|Vaughan|1950}}<blockquote>Suppose that each of a (possibly infinite) set of boys is acquainted with a finite set of girls. Under what conditions is it possible for each boy to marry one of his acquaintances? It is clearly necessary that every finite set of ''k'' boys be. collectively, acquainted with at least ''k'' girls... this condition is also sufficient.</blockquote>