Let $X$ be a set with more than two elements. Define a relation $R$ on $P (X)$, the power set of X, by $(A, B) \in R$ if and only if $A \subseteq B$. Show that $R$ is a partial order on $P (X)$. Is it a well ordering? Is it a total ordering?
So there is a set $X$, that has more than 2 elements. The power set of $X$ has $(A,B)\in R$ if and only if $A \subseteq B$. How would I show if it is partial order on $P(X)$? Or if it is well and/or total ordering? If I am correct, partial ordering is when reflexive, anti symmetric, and transitive. I am not to sure about total ordering. Well ordering = a set having a least element, if I am not mistaken.