Let S be a finite set and let S
1, S
2, …, St
be subsets of S, not necessarily distinct. Does there exist a set of distinct representatives (SDR) for S
1, S
2, …, St
? That is, does there exist a subset {a
1, a
2, …, at
} of S such that ai
∊ Si
, 1 ≦ i ≦ t, and ai
≠ aj
if i ≠ j? The following theorem of Hall [2; 6, p. 48] gives the answer.
THEOREM. The subsets S
1, S
2, …, St have an SDR if and only if for each s, 1 ≦ s ≦ t, |S
i
1
∪ S
i
1
∪ … ∪ Sis
| ≧ s for each s-comhination {i
1, i
2, …, is
} of the integers 1, 2, …, t.
(Here and below, |A| denotes the number of elements in A.)