In each of Exercises 1–2, you are given a directed graph for a relation on the set \(A = \{a,b,c,d\}\text{.}\) Determine whether the relation is a partial order. Justify your answers.
1.
2.
Testing partial orders.
In each of Exercises 3–6, you are given a set \(A\) and a relation \(R\) on \(A\text{.}\) Determine whether the relation is a partial order. Justify your answers.
3.
\(A\) is the set of all Augustana students; \(a \mathrel{R} b\) means that student \(a\) has a higher GPA than student \(b\text{.}\)
4.
\(A\) is the power set of some finite set; \(S \mathrel{R} T\) means \(\card{S} \le \card{T}\text{.}\)
5.
\(A\) is the set of words on some alphabet; \(w \mathrel{R} w'\) means \(\length{w} \le \length{w'}\text{,}\) where \(\length{w}\) means the length of word \(w\text{.}\)
6.
\(A = \R \cartprod \R\text{;}\)\((x1,y1) \mathrel{R} (x2,y2)\) means \(x_1 \le x2\) and \(y_1 \le y2\text{.}\)
Drawing Hasse diagrams.
In each of Exercises 7–8, you are given a finite, partially ordered set \(A\text{.}\) Draw the Hasse diagram.
7.
\(A = \powset{\{1,2,3,4\}}\) under the subset relation.
8.
\(A = \words{\Sigma}_4\text{,}\) the set of words of length \(4\) in the alphabet \(\Sigma = \{0,1\}\text{,}\) under dictionary order.
9.
Draw all possible valid Hasse diagrams for each of the sets \(A = \{a,b\}\) and \(B = \{a,b,c\}\text{.}\) (Thus, you will have determined all possible partial orders on those sets.)
10.
Consider the “divides” relation \(\mathord{\mid}\) on \(\posset{\N}\text{.}\) Provide an example of a set \(A \subseteq \posset{\N}\)
(a)
that is finite, and on which \(\mathord{\mid}\) is a total order.
(b)
that is infinite, and on which \(\mathord{\mid}\) is a total order.
(c)
on which \(\mathord{\mid}\) is a partial order but not a total order.
11.
Let \(A = \{0,1,2\}\text{,}\) and consider the partial order \(\mathord{\subseteq}\) on the power set \(\powset{A}\text{.}\) List all pairs of incomparable elements in \(\powset{A}\text{.}\)
In each of Exercises 12–16, you are given a partially ordered set \(A\text{.}\) Determine any and all maximal, maximum, minimal, and minimum elements.
12.
\(A = \posset{\N}\) under the usual \(\mathord{\le}\text{.}\)
13.
\(A = \posset{\Q}\) under the usual \(\mathord{\le}\text{.}\)
14.
\(A = \N \relcmplmnt \{0,1\}\) under the “divides” relation \(\mathord{\mid}\text{.}\)
15.
\(A = \{2,5,11,13,22,65,110,143,496\}\) under the “divides” relation \(\mathord{\mid}\text{.}\)
16.
\(A\) is the set of the first ten prime numbers under the “divides” relation \(\mathord{\mid}\text{.}\)
17.
Suppose \(\mathord{\partorder}\) 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 \(\mathord{\partorder}\text{?}\)
