The exercise is to find sub-orders of $(P( \lbrace x,y,z \rbrace ), \subset)$ such they are: (1)Total ordered, (2) partial ordered and not total , (3) partial but not well ordered.
First of all , $P(\lbrace x,y,z \rbrace)= \lbrace \emptyset ,\lbrace x\rbrace ,\lbrace y \rbrace , \lbrace z \rbrace ,\lbrace x,y\rbrace ,\lbrace x,z \rbrace ,\lbrace y,z \rbrace ,A \rbrace$
So , by a total order I thought of $\lbrace\lbrace x \rbrace , \lbrace x,y \rbrace , \lbrace x,y,z \rbrace \rbrace$ since it is antysimetric, transitive and total . For a partial order but not total o thought of $\lbrace \lbrace x \rbrace \lbrace y \rbrace \lbrace x,y \rbrace , \lbrace x,y,z \rbrace \rbrace $ since it is partial but we can not compare $\lbrace x \rbrace$ and $\lbrace y \rbrace $ . Does my examples are right? how do I show a partial order that is not well ordered?? Thanks