Counting by Fibres, Double Counting and the Pigeonhole Principle

In the setting of Finite Combinatorics: Notation, let $A$ and $B$ be finite sets and $f:A\to B$ a map.

(i) Counting by fibres.

$$\#A=\sum_{y\in B}\#f^{-1}\{y\}.$$

(ii) Double counting. Let $R\subseteq A\times B$, and for $a\in A$ and $b\in B$ let $R_{a}=\{y\in B:(a,y)\in R\}$ and $R'_{b}=\{x\in A:(x,b)\in R\}$. Then

$$\sum_{a\in A}\#R_{a}=\#R=\sum_{b\in B}\#R'_{b}.$$

(iii) Pigeonhole principle. If $k\in\mathbb{N}_{0}$ and $\#A>k\cdot\#B$, then there is $y\in B$ with $\#f^{-1}\{y\}\ge k+1$.

(iv) Fibres of equal size. If $r\in\mathbb{N}_{0}$ and $\#f^{-1}\{y\}=r$ for every $y\in B$, then

$$\#A=r\cdot\#B.$$