pigeonhole principle

1. If $n$ objects are distributed over $m$ places, and if $n>m$, then some place receives at least two objects.
2. If $n$ objects are distributed over $n$ places such that no place receives more than one object, then each place receives exactly one object.
3. If $S$ and $T$ are sets, and the cardinality of $S$ is greater than the cardinality of $T$, then there is no injective function from $S$ to $T$.
4. If $n$ objects are distributed over $m$ places, and if $n<m$, then some place receives no object.
5. If $n$ objects are distributed over $n$ places such that no place receives no object, then each place receives exactly one object.
6. If $S$ and $T$ are sets, and the cardinality of $S$ is less than the cardinality of $T$, then there is no surjective function from $S$ to $T$.

Formally, "there does not exist an injective function whose codomain is smaller than its domain."
Equivalently, "there does not exist a surjective function whose domain is smaller than its codomain."