Let (S, d) and (S, d) be two partially ordered sets. (S,
Chapter 8, Problem 19(choose chapter or problem)
Let \((S, \preceq)\) and \((S^\prime, \prec^\prime)\) be two partially ordered sets. \((S, \preceq)\) is isomorphic to \((S^\prime, \preceq^\prime)\) if there is a bijection \(f: S \rightarrow S^\prime\) such that for x, y in \(S, x \prec y \rightarrow f(x) \prec^\prime f(y)\) and \(f(x) \prec^\prime f(y) \rightarrow x \prec y\).
a. Show that there are exactly two nonisomorphic, partially ordered sets with two elements (use diagrams).
b. Show that there are exactly five nonisomorphic, partially ordered sets with three elements.
c. How many nonisomorphic, partially ordered sets with four elements are there?
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