Proof
(related to Lemma: Composition of Relations Preserves Their Right-Uniqueness Property)
Let \(A,B,C\) be sets and let let \(R_1\subseteq A\times B\) and \(R_2\subseteq B\times C\) be two right-unique relations.
Case \((1)\)
- Suppose, there exist at least one tripple $(a,b,c)$ with $a\in A,$ $b\in B,$ and $c\in C$ such that $aR_1b$ and $bR_2c$.
- From the right-uniqueness of $R_1$ it follows that $b$ is uniquely determined by $a$.
- From the ritht-uniqueness of $R_2$ it follows that $c$ is uniquely determined by $b$.
- Alltogether, $c$ is uniquely determined by $a$.
- From the definition of composition of relations, it follows that $(R_2\circ R_1)\subseteq A\times C$ is right-unique.
Case \((2)\):
- Now suppose that such a tripple $(a,b,c)$ does not exist.
- Then the lemma is vacuously true.
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References
Bibliography
- Knauer Ulrich: "Diskrete Strukturen - kurz gefasst", Spektrum Akademischer Verlag, 2001
