Discussion and Examples
Example 1: Consider S = {A1, A2, A3} with
- A1 = {1, 2, 3}
- A2 = {1, 4, 5}
- A3 = {3, 5}.
A valid SDR would be {1, 4, 5}. (Note this is not unique: {2, 1, 3} works equally well, for example.)
Example 2: Consider S = {A1, A2, A3, A4} with
- A1 = {2, 3, 4, 5}
- A2 = {4, 5}
- A3 = {5}
- A4 = {4}.
No valid SDR exists; the marriage condition is violated as is shown by the subcollection {A2, A3, A4}.
Example 3: Consider S= {A1, A2, A3, A4} with
- A1 = {a, b, c}
- A2 = {b, d}
- A3 = {a, b, d}
- A4 = {b, d}.
The only valid SDR's are (c, b, a, d) and (c, d, a, b).
Read more about this topic: Hall's Marriage Theorem
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