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I am having trouble finding an equivalence relation, $R$, on the set $\{1,2,3,4\}$.

I am given that $(1,1), (1,2), (2,3) \in R$ but $R\ne A \times A$.

I'm not necessarly looking for the answer just what method one would take? Thanks!

dfeuer
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Adam
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  • Equality is always an equivalence relation. Are you looking for a non trivial one? Does $<$ work? – TheNumber23 Oct 13 '13 at 18:00
  • I guess you're looking for an equivalence relation $R$ on $A \times A$ that contains the three specified pairs, but is not $A \times A$ itself. Is that correct? If so, can you update your question to make this clear? – Magdiragdag Oct 13 '13 at 18:02
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    Have you tried taking the transitive reflexive closure of the symmetric closure of ${(1,1),(1,2),(2,3)}$? – dfeuer Oct 13 '13 at 18:06
  • thanks dfeuer, yes that's it! – Adam Oct 13 '13 at 18:26

2 Answers2

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If you are given some elements of $R \subset A\times A$, you should probably start out by expanding those equivalences in order to meet the requirements of an equivalence relation; e.g. if $(1,2) \in R$, then $(2,1)\in R$; if $(1,2), (2,3) \in R$, then $(1,3) \in R$, etc. Doing this until you get a valid equivalence relation will give you the minimal equivalence relation $R \subset A\times A$ that has the given elements in it.

Epsilonaut
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Hint: By the givens one equivalence class contains at least $1$, $2$, and $3$. If it contained $4$ as well, ...

Hagen von Eitzen
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