Establish an equivalence relation on the finite set of prime integers.

Question

Given the finite set of consecutive prime integers:

{2, 3, 5, 7, 11}

It is required to establish an equivalence relation on the set.

Answer 1

Establishing an equivalence relation on a set $A$ comes to the same as establishing a partition of $A$.

If $\mathcal P$ denotes a partition of $A$ then the corresponding equivalence relation is: $$aRb\text{ if and only if }a,b\in P\text{ for some }P\in\mathcal P$$

So just choose some partition of $\{2,3,5,7,11\}$ and define $R$ as above.

If conversely $R$ denotes an equivalence relation on $A$ then the corresponding partition is the set of equivalence classes of $R$.

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Answer 2

Thanks to the comment by arberavdullahu, the relation $aRb$ iff $p|(a-b)$, where $p$ denotes any prime in the set defines an equivalence relation on the set. Indeed,

1)$p|(a-a)=0\implies R $ is reflexive

2)$p|(a-b)\implies p|(b-a)=-(a-b)\implies R$ is symmetric

3)$p|(a-b) $ and $p|(b-c)\implies p|(a-c)=(a-b)+(b-c)\implies R$ is transitive.