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:
“In the order of literature, as in others, there is no act that is not the coronation of an infinite series of causes and the source of an infinite series of effects.”
—Jorge Luis Borges (1899–1986)
“To the extent to which genius can be conjoined with a merely good human being, Haydn possessed genius. He never exceeds the limits that morality sets for the intellect; he only composes music which has “no past.””
—Friedrich Nietzsche (1844–1900)