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
Famous quotes containing the words marriage, condition and/or extend:
“In mid-life the man wants to see how irresistible he still is to younger women. How they turn their hearts to stone and more or less commit a murder of their marriage I just don’t know, but they do.”
—Patricia Neal (b. 1926)
“The Germans are always too late. They are late, like music, which is always the last of the arts to express a world condition,—when that world condition is already in its final stages. They are abstract and mystical.”
—Thomas Mann (1875–1955)
“At any rate, the line is
Long and electric. Lean beyond and nod.
Be sprightly. Wave. Extend your hand and teeth.”
—Gwendolyn Brooks (b. 1917)