# The Pigeonhole Principle

Last updated: Oct. 21, 2016

According to the Pigeonhole Principle, if you have more pigeons than pigeonholes (as in photo), then at least two pigeons will be in the same hole. Here we present a *non-numeric* version.

Usually, *more* pigeons than pigeonholes is taken to mean that the *number* of pigeons is *greater than* the *number* of pigeonholes. Here, we take more pigeons than pigeonholes to mean that the set of pigeonholes cannot be mapped *onto* the set the pigeons.

In this sense then, we can prove that, if we put a non-empty set of pigeons into a set of pigeonholes and there are more pigeons than holes, then we will have put at least two pigeons in the same hole. Note that there is no requirement here that there be finitely many pigeons or pigeonholes.

See formal proof (70 lines) at The Pigeonhole Principle: A non-numeric version.