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:
“I feel that the Godhead is broken up like the bread at the Supper, and that we are the pieces. Hence this infinite fraternity of feeling.”
—Herman Melville (1819–1891)
“I think middle-age is the best time, if we can escape the fatty degeneration of the conscience which often sets in at about fifty.”
—W.R. (William Ralph)