03-00-AnInjectionBetweenTwoFiniteSetsOfTheSameCardinalityIsBijective.tex

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\pmtitle{an injection between two finite sets of the same cardinality is bijective}
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\pmtype{Theorem}
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%\pmkeywords{bijective}
%\pmkeywords{injective}
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\begin{document}
\begin{lemma}
Let $A,B$ be two finite sets of the same cardinality. If $f\colon A \to B$ is an injective function then $f$ is bijective.
\end{lemma}
\begin{proof}
In order to prove the lemma, it suffices to show that if $f$ is an injection then the cardinality of $f(A)$ and $A$ are equal. We prove this by induction on $n=\text{card}(A)$. The case $n=1$ is trivial. Assume that the lemma is true for sets of cardinality $n$ and let $A$ be a set of cardinality $n+1$. Let $a\in A$ so that $A_1=A-\{a\}$ has cardinality $n$. Thus, $f(A_1)$ has cardinality $n$ by the induction hypothesis. Moreover, $f(a)\notin f(A_1)$ because $a\notin A_1$ and $f$ is injective. Therefore:
$$f(A)=f(\{a\}\cup A_1)=\{f(a)\}\cup f(A_1)$$
and the set $\{f(a)\}\cup f(A_1)$ has cardinality $1+n$, as desired.
\end{proof}
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