Lets say $R={(1,3),(2,2),(1,4)}⊆\underline{4}×\underline{4}$ is a relation to $\underline{4}$. I'm looking for the smallest equivalent relation to $\underline{4}$.
My idea was the following: It has to be reflexive, so we need $(1,1),(3,3),(4,4)$
It has to be transitive: We don’t need more relations
It has to be symmetric: We need $(3,1),(4,1)$
Now I wanted to ask you, if this is right, because I think, now I could create a relation $(4,1)∧(1,3)$ which doesn’t imply $(4,3)$, which would be contradict to transitive. Does this mean that I also have to add the relation (4,3) or is it an equivalent relation without? Do I have to go over the 3 steps over and over again, until there is nothing more, I miss out like the relation $(4,3)$?
You have to add $(4,3)$ and $(3,4)$ and you are done.
Notice that $2$ is only related to itself.
If you draw out the graph with $4$ nodes and introduce an arc if they are related, you will see a complete graph of $3$ nodes and another complete graph of a single node (that would be node $2$).