if $S$ are disconnected on his subspace topology then there exists $U \cap S$ where $U$ are open on $X$ and $H\cup S$ where $H$ are open in $X$ that $S=(U\cap S)\cup (H\cap S) = (H\cap U)\cup S$ but by this how can i find the open sets in $X$ such that $S$ is his union?
$S$ disconnected on $X$ means $S$ is disconnected in its subspace topology by defination! Your question makes no sense
$S = (H\cup U)\cap S $ implies $S \subseteq H \cup U$. You seem to be implying there are two definitions of a disconnected subspace, but there is only one: $S$ is disconnected in its subspace topology.