There exists a system of distinct representatives for a family of sets , , ..., iff the union of any of these sets contains at least elements for all from 1 to (Harary 1994, p. 53).
Hall's Theorem
See also
Hall's ConditionExplore with Wolfram|Alpha
References
Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994.Referenced on Wolfram|Alpha
Hall's TheoremCite this as:
Weisstein, Eric W. "Hall's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HallsTheorem.html