For this exercise, be careful to use only the definitions and not your intuitions that are familiar to you about the symbols \(\leq\) and \(<\). Consider a set \(S\) with a binary relation \(<\) that is transitive and fulfills for all \(x\) in \(S\)

\[\begin{equation*} \text { not } x

Define the relation \(\leq\) on \(S\) by (1.7), that is, \(x \leq y \Leftrightarrow x

(a) Show that \(\leq\) is a partial order, and that \(<\) is obtained from \(\leq\) via (1.6), that is, \(x

(b) Show that if \(<\) is obtained from a partial order \(\leq\) via (1.6), then \(<\) is transitive and irreflexive, and (1.7) holds, that is, \(x \leq y \Leftrightarrow x