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Hall's marriage theorem
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=== Examples === [[File:Halls theorem positive example2.svg|thumb|upright=1.25|example 1, marriage condition met]] ;Example 1 : Consider the family <math>\mathcal F=\{A_1,A_2,A_3\}</math> with <math>X=\{1,2,3,4,5\}</math> and <math display=block> \begin{align} A_1&=\{1,2,3\}\\ A_2&=\{1,4,5\}\\ A_3&=\{3,5\}.\\ \end{align} </math> The transversal <math>\{1,3,5\}</math> could be generated by the function that maps <math>A_1</math> to <math>1</math>, <math>A_2</math> to <math>5</math>, and <math>A_3</math> to <math>3</math>, or alternatively by the function that maps <math>A_1</math> to <math>3</math>, <math>A_2</math> to <math>1</math>, and <math>A_3</math> to <math>5</math>. There are other transversals, such as <math>\{1,2,3\}</math> and <math>\{1,4,5\}</math>. Because this family has at least one transversal, the marriage condition is met. Every subfamily of <math>\mathcal F</math> has equal size to the set of representatives it is mapped to, which is less than or equal to the size of the union of the subfamily. [[File:Halls theorem negartive example2.svg|thumb|upright=1.25|example 2, marriage condition violated]] ;Example 2 : Consider <math>\mathcal F=\{A_1,A_2,A_3,A_4\}</math> with <math display=block> \begin{align} A_1&=\{2,3,4,5\}\\ A_2&=\{4,5\}\\ A_3&=\{5\}\\ A_4&=\{4\}.\\ \end{align} </math> No valid transversal exists; the marriage condition is violated as is shown by the subfamily <math>\mathcal G=\{A_2,A_3,A_4\}</math>. Here the number of sets in the subfamily is <math>|\mathcal G|=3</math>, while the union of the three sets <math>A_2\cup A_3\cup A_4=\{4,5\}</math> contains only two elements. A lower bound on the different number of transversals that a given finite family <math>\mathcal F</math> of size <math>n</math> may have is obtained as follows: If each of the sets in <math>\mathcal F</math> has cardinality <math>\geq r</math>, then the number of different transversals for <math>\mathcal F</math> is either <math>r!</math> if <math>r\leq n</math>, or <math>r(r-1)\cdots(r-n+1)</math> if <math>r>n</math>.<ref>{{harvnb|Reichmeider|1984}}, p.90</ref> Recall that a transversal for a family <math>\mathcal F</math> is an ordered sequence, so two different transversals could have exactly the same elements. For instance, the collection <math>A_{1}=\{1,2,3\}</math>, <math>A_{2}=\{1,2,5\}</math> has <math>(1, 2)</math> and <math>(2, 1)</math> as distinct transversals.
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