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