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 consciousness in each man is a sliding scale, which identifies him now with the First Cause, and now with the flesh of his body; life above life, in infinite degrees.”
—Ralph Waldo Emerson (1803–1882)
“Certain anthropologists hold that man, having discovered tools, ceased to evolve biologically. Animals, never having discovered them, continue to fashion drills out of their beaks, oars out of their hind feet, wings out of their forefeet, suits of armor out of their hides, levers out of their horns, saws out of their teeth. Whether this be true or not, all authorities agree that man is the tool-using animal. It sets him off from the rest of the animal kingdom as drastically as does speech.”
—Stuart Chase (1888–1985)