Pigeonhole principle

Statement

Let $A$ and $B$ be finite sets with $|A| > |B|$, and let $f : A \to B$ be a function. Then there are elements $a \neq a'$ of $A$ with $f(a) = f(a')$. In other words, $f$ is not injective.

Remarks

In the usual image, if $n + 1$ pigeons sit in $n$ holes, then some hole has at least two pigeons. Example: among any $13$ people, two have their birthdays in the same month. The generalized pigeonhole principle gives a fibre with more than $n$ elements. The infinite pigeonhole principle is a version for infinite sets.

Explanations

L1 human

If there are more objects than boxes, some box has two objects. For example, in any group of 367 people, two share a birthday, because a year has at most 366 days.