Hall's Marriage Theorem - Discussion and Examples

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).

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