Find sub-orders of $(P( \lbrace x,y,z \rbrace ), \subset)$ such they are: Total ordered, partial ordered and not total , 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 a \rbrace \lbrace b \rbrace \lbrace x,y \rbrace , \lbrace x,y,z \rbrace \rbrace $ since it is partial but we can not compare $\lbrace a \rbrace$ and $\lbrace b \rbrace $ . Does my examples are right? how do I show a partial order that is not well ordered?? Thanks
Your examples seems right! A set $X$ is well ordered if every non empty set $B$ of $X$ possesses a smallest element. So, a set $Y$ is not well ordered if there is some nonempty subset $Z$ of $Y$ which has no smallest element.
Ok, consider $X = \left\{\{x\},\{y\},\{x,\ y\},\ \{x,\ y,\ z\}\right\}$---A partial order (not total though) and let $B = \left\{ \{x\},\ \{y\},\ \{x,\ y,\ z\}\right\}$. It looks like $B$ is a non empty subset of $X$ with no smallest element---so $B$ would not be well ordered. By the way, if a set is partially ordered, then any subset of it is partially ordered---the induced partial order. $B$ has bottom elements (minimal elements) namely $\{x\}$ and $\{y\}$ so which is the smallest? Bear in mind that these two are incomparable---hence, no smallest one.