Answer to Question #146729 in Discrete Mathematics for Promise Omiponle

Question

Suppose S is a set containing 5 elements, and that ⪯ is a total ordering of S. Draw the Hasse diagram for ⪯ (no need to label the vertices in your diagram).

Answer 1

$S=\{a,b,c,d,e\}$

$\preceq$ is a total order on S.

Any total order $\preceq$ is reflexive, antisymmetric, transitive. Therefore (S,$\preceq)$ form a poset.

Again as $\preceq$ is a total order any two elements of S are comparable. Therefore

we can order the elements of S in such a way so that they form a chain

$a\preceq b \preceq c\preceq d \preceq e$ (say).

The corresponding Hasse diagram is drawn in this convention using the undirected line, the $\preceq$ relation (hence, the ordering of the elements) is read from the bottom up.