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The question starts with this:

Given the following relation S on Z×Z where Z= {a, b, c, d, e}: S={(a, a),(b, b),(a, b),(b, a),(c, c),(d, d),(e, e),(c, e),(d, e),(e, c),(e, d)}

Then it asks to see if it is an equivalence relation.

I think I'm missing something because doesn't Z x Z =

{(a,a),(a,b),(a,c),(a,d),(a,e),(b,a),(b,b),(b,c),(b,d),(b,e),(c,a),(c,b),(c,c),(c,d),(c,e),(d,a),(d,b),(d,c),(d,d),(d,e),(e,a),(e,b),(e,c),(e,d),(e,e)}

tinkertailor
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    Do you know what an equivalence relation is? – Sayan Dutta Aug 05 '21 at 16:11
  • Note: technically, $S$ is a binary relation on $Z$, not $Z \times Z$. – user2628206 Aug 05 '21 at 16:14
  • A relation $S$ "over Z" is merely a subset of $Z\times Z$. Your example here... indeed, $S$ is a subset of $Z\times Z$ and so is a valid relation over $Z$. The question is whether or not your relation $S$ satisfies certain "nice" properties that often interest us about relations – JMoravitz Aug 05 '21 at 16:15
  • If your complaint is about the phrasing, yes... I would have said $S$ is a relation over $Z$. It is correct that $S$ is not a subset of $(Z\times Z)\times (Z\times Z)$ and so should not be called "a relation over $Z\times Z$" but it should be clear from context that we intended $S$ to be treated as a relation over $Z$. – JMoravitz Aug 05 '21 at 16:17
  • Agreed, but the poster may not understand the subtlety – user2628206 Aug 05 '21 at 16:18
  • Yea sorry guys it wasn't that deep. I actually didn't understand something basic about this. I didn't realize that S was a subset. I thought it meant literally Z X Z = S – tinkertailor Aug 05 '21 at 16:24

2 Answers2

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A relation $R$ on a set $T$ is a subset of $T \times T$. Hence the problem in the question totally makes sense as $S$ is indeed a subset of $Z \times Z$.

You're just asked to validate that your relation $S$ satisfies the axioms of an equivalence relation. Namely: reflexivity, symmetry and transitivity.

mathcounterexamples.net
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Question is whether the relation $S$ is reflexive, symmetric, and transitive.

S={(a, a),(b, b),(a, b),(b, a),(c, c),(d, d),(e, e),(c, e),(d, e),(e, c),(e, d)}

Its reflexive and symmetric but not transitive, since $(c,e), (e,d)\in S$ but $(c,d)\not\in S$.

Wuestenfux
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