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:
“The universe is then one, infinite, immobile.... It is not capable of comprehension and therefore is endless and limitless, and to that extent infinite and indeterminable, and consequently immobile.”
—Giordano Bruno (1548–1600)
“There is a small steam engine in his brain which not only sets the cerebral mass in motion, but keeps the owner in hot water.”
—Unknown. New York Weekly Mirror (July 5, 1845)