Marriage Condition Does Not Extend
The following example, due to Marshall Hall, Jr., shows that the marriage condition (MC) will not guarantee the existence of an SDR in an infinite family in which infinite sets are allowed.
Let S be the family, A0 = {1, 2, 3, ...}, A1 = {1}, A2 = {2}, ..., Ai = {i}, ...
The marriage condition (MC) holds for this infinite family, but no SDR can be constructed.
The more general problem of selecting a (not necessarily distinct) element from each of a collection of sets (without restriction as to the number of sets or the size of the sets) is permitted in general only if the axiom of choice is accepted.
Read more about this topic: Hall's Marriage Theorem
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