Infinite Sets
The pigeonhole principle can be extended to infinite sets by phrasing it in terms of cardinal numbers: if the cardinality of set A is greater than the cardinality of set B, then there is no injection from A to B. However in this form the principle is tautological, since the meaning of the statement that the cardinality of set A is greater than the cardinality of set B is exactly that there is no injective map from A to B. What makes the situation of finite sets interesting is that adding at least one element to a set is sufficient to ensure that the cardinality increases.
Read more about this topic: Pigeonhole Principle
Famous quotes containing the words infinite and/or sets:
“Nothing could his enemies do but it rebounded to his infinite advantage,—that is, to the advantage of his cause.... No theatrical manager could have arranged things so wisely to give effect to his behavior and words. And who, think you, was the manager? Who placed the slave-woman and her child, whom he stooped to kiss for a symbol, between his prison and the gallows?”
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