$P$ be a poset and $X, Y \subseteq P$. Is it true that if $\downarrow X = \downarrow Y$ , then $X= Y$?

Question

Let $P$ be a poset and $X, Y \subseteq P$. Is it true that if $\downarrow X = \downarrow Y$ (the generated down-sets) then $X= Y$?

I must show that when $\forall x \in X \,\,\exists \,\, y\in Y$ s.t $x\le y\,\,\,$ and $\,\,\,\forall y \in Y \,\, \exists \,\, x \in X\,\,$ s.t $\,\,y\le x\,\,\,$ this implies $X=Y$. Because the first condition gives me $\downarrow X \subseteq \downarrow Y$ and the second one results in $\downarrow Y \subseteq \downarrow X$

Answer 1

No. For instance, if $X$ is any set that is not itself a down-set, then you could take $Y={\downarrow X}$, and then ${\downarrow Y}=Y={\downarrow X}$.