Let $1\leq n,r$ be integers and $f:[1,n]\rightarrow[1,r]$ a mapping. For $1\leq j\leq r$, let $\epsilon_j:=|f^{-1}(j)|$. I want to show that $$|f^{-1}(\epsilon_j)|=\epsilon_j$$ for all $1\leq j\leq r$. It seems like it might hold but I'm having difficulty proving/disproving it. Any suggestions?
2026-04-01 09:37:52.1775036272
Cardinality of preimages of finite mappings
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Any non surjective function has a $j$ such that $\epsilon _j=0,$ the inverse is not even defined there. Another example would be just $f=1\mapsto 2, 2\mapsto 2,3\mapsto 3$
$$|f^{-1}(3)|=1,|f^{-1}(1)|=0.$$ It seems really strong your condition. Why you think is true?