Hall's Theorem

Hall's Theorem provides a condition for when it's possible to pair every member of one group with a unique member of another group, such as matching students to projects or workers to tasks. It states that such a perfect matching exists if and only if, for any subset of the first group, the number of potential matches in the second group is at least as large as the number of members in that subset. This ensures there are enough suitable options for each individual in the initial group, guaranteeing that a complete, one-to-one pairing is possible.