Any total order on $X$ is maximal element

Question

Let $X$ be a non-empty set and $R$ be the set of partial orders on $X$.

(1) Show that $R$ is partially ordered by inclusion $\subset$.

(2) Show that any total order on $X$ is a maximal element in $(R, \subset)$

Answer 1

HINT: Suppose that $R_1$ is a total order, and $R_2$ extends it, then for some $(x,y)\in R_2$ we have that $(x,y)\notin R_1$. Use the fact that $R_1$ is total, and arrive at a contradiction to the antisymmetry of $R_2$.