The Pigeonhole Principle

Ten pigeons in nine holes

Last updated: June 5, 2022

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 at The Pigeonhole Principle: A non-numeric version.