Let \(F \subseteq \N\) represent the set of all divisors of \(30\text{.}\) Let \(A = \{a,b,c\}\text{.}\)
Note: In Task c you will compare your work from Task a and Task a, so keep your work!
(a)
Draw the Hasse diagram for the subset partial order \(\mathord{\subseteq}\) on \(\powset{A}\text{.}\)
(b)
Draw the Hasse diagram for the “divides” partial order \(\mathord{\mid}\) on \(F\text{.}\)
(c)
Compare your two Hasse diagrams. Can you devise a function \(\funcdef{f}{F}{\powset{A}}\) that would deserve to be called an order-preserving correspondence between \(F\) and \(\powset{A}\text{?}\)
Activity19.2.
Suppose \(\mathord{\partorder}\) is a partial order on a set \(A\text{.}\) Verify that the inverse relation\(\inv{\partorder}\) is also a partial order on \(A\) by verifying that it is reflexive, antisymmetric, and transitive.
Activity19.3.
Let \(A = \{a,b,c,d,e\}\text{.}\) Carry out the following steps for each of the scenarios below.
Draw the Hasse diagram for a partial order on \(A\) with the requested features.
In your diagram, identify all maximal/minimal elements.
Identify all pairs of incomparable elements.
(a)
\(A\) has both a maximum and a minimum.
(b)
\(A\) has a maximum but no minimum.
(c)
\(A\) has a minimum but no maximum.
(d)
\(A\) has neither a maximum nor a minimum.
Activity19.4.
Suppose \(\mathord{\preceq}\) is a partial order on the set \(A = \{0,1,2\}\) such that \(1\) is a maximal element. What are the possibilities for the Hasse diagram of \(\mathord{\preceq}\text{?}\)
Activity19.5.
Using the proper strategy for proving uniqueness (see Procedure 6.10.2), prove that if a partially ordered set \(A\) has a maximum element, then that element is the unique maximum element.
How can your proof be modified to show that a minimum element is also unique?
Activity19.6.
Recall that \((a,b)\subseteq \R\) means an open interval on the real number line:
Let \(\mathord{\le}\) be the usual “less than or equal to” total order on the set
\begin{equation*}
A = (-2,0)\cup(0,2) \text{.}
\end{equation*}
Consider the subset
\begin{equation*}
B = \setdef{-\frac{1}{n}}{n \in \N, \, n \ge 1} \subseteq A \text{.}
\end{equation*}
Determine an upper bound for \(B\) in \(A\text{.}\) Then formally prove that \(B\) has no least upper bound in \(A\) by arguing that every element of \(A\) fails the criteria in the definition of least upper bound.
