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Hello I have a question about possible equivalence relations.

I know that a relation can be Reflexive, Symmetric , Transitive.

But my question is, is there any strict limitations one has on the other.

For example if we had a relation then there are eight possible combinations of the above, for example we could have R S T and not R , S, T, or Not R, Not S, T, for example.

To me they all seem possible except for a relation that is not reflexive but symmetric and transitive.

Any insight?

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  • $\begingroup$ All eight are possible. The empty relation for example satisfies being not reflexive while being both symmetric and transitive. $\endgroup$ – JMoravitz Oct 17 '15 at 19:57
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What about a relation on a set where nothing relates to anything? This is symmetric and transitive, but not reflexive.

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  • $\begingroup$ Hm interesting, but how would I write such as a subset of the Cartesian product? $\endgroup$ – PersonaA Oct 17 '15 at 20:00
  • $\begingroup$ Like this: $\emptyset$. $\endgroup$ – Alex S Oct 17 '15 at 20:00
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    $\begingroup$ @PersonaA There are more examples beyond the empty relation as well. So long as there is at least one element $x$ which is not related to anything (especially itself), even if the rest of the relation satisfies symmetry and transitivity, since $x$ is not related to itself, it will not be reflexive. For example $R\subseteq \{1,2,3\}\times \{1,2,3\}$ with $R=\{(1,1),(1,2),(2,1),(2,2)\}$ $\endgroup$ – JMoravitz Oct 17 '15 at 20:04

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