Equivalence class for non-equivalence relations

Question

I have a homogeneuos relation $R\subseteq X\times Y,\,\, X =Y$ that is

1. symmetric
2. reflexive
3. not transitive

What is $[x]_R = \{y:(x,y)\in R\}$ called and what is the proper notation if not $[x]_R$?

I have another relation $S\subseteq X\times Y,X=Y$ that is

1. antisymmetric
2. reflexive
3. transitive

What is the transitive closure of $S$ given x called and denoted? (If you know what I mean...)

Answer 1

We have that $R[A]=\{y \mid \exists x\in A,(x,y)\in R \}$ is called the image of a set $A$ under a relation $R$.

Thus, we have $R[ \{ x \} ]$ and we can "simplify" it to $R[x]$.

Sometimes also used: $R‘‘A$; in this case we may write $R‘‘ \{ x \}$.