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 nature’s infinite book of secrecy
A little I can read.”
—William Shakespeare (1564–1616)
“bars of that strange speech
In which each sound sets out to seek each other,
Murders its own father, marries its own mother,
And ends as one grand transcendental vowel.”
—Randall Jarrell (1914–1965)