Equivalence Relations assistance needed

Question

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!

Answer 1

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

Answer 2

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.