Originally Posted by

**Ester** I have this kind of exercise:

Let < be the partial order in the set A and define function $\displaystyle F:A \rightarrow P(A)$ by determining $\displaystyle F(a) = \{x \in A \mid x \leq a\}$, when $\displaystyle a \in A$.

Let $\displaystyle S = ran(F)$. Show that F is the isomorphism between structures $\displaystyle <A, < >$ and $\displaystyle <S, \subset >$.

Okey, my problem is how to mark function between A and S. I think I can't do it like $\displaystyle F:A\rightarrow S$, because I have already defined F before.

Rest of the exercise I can handle, I just have to show that the function between A and S is bijection and that $\displaystyle \forall x, y \in A(x <_A y \leftrightarrow "function"(x) \subset "function(y)")$.