finding equivalence class of a set $U$

Question

Let $R$ be an equivalence relation on a non empty set $X=[0, 1].$ $x R y$ iff $x=y$ or $x, y \in \{0, 1\}$.

Define $R[A]=\{y \in X: x R y ,\text{for some $x \in A$} \}$

Then what is $R(U)?$, where $U=(0, 1/2]$

my ans is $R(U)=U$ but the given ans is $\{0\}\cup U$

Answer 1

You are correct.

$\operatorname R{(0;1/2]}~{=\{y\in[0;1]: \exists x\in(0;1/2]~(x\operatorname R y)\}\\=\{y\in[0;1]: \exists x\in(0;1/2]~(x=y\vee(x\in\{0,1\}\wedge y\in\{0,1\}))\}\\=\{y\in[0;1]: \exists x\in(0;1/2]~(x=y)~\vee~\exists x\in(0;1/2]\cap \{0,1\}~(y\in\{0,1\})\}\\=\{y\in[0;1]: \exists x\in(0;1/2]~(x=y)~\vee~\bot\}\\=\{y\in [0;1]\cap (0;1/2]\}\\=(0;1/2]}$