Another way to define the elements of the relation is a Hasse diagram. FigureΒ 4.4.2 shows the Hasse diagram for \(R.\) The rules for constructing these diagrams are in ListΒ 4.4.3.
Note that constructing a Hasse diagram requires sorting the elements of the set. Each level of the diagram in some sense is βbeforeβ the levels above it. Because of this ordering in posets we write \(a \preceq b\) to mean \(aRb\text{.}\)
Determine whether the following relation is a poset. Let the relation \(R\) on the set of numbers \(\{1,2,3,6\}\) be defined by \(aRb\) if and only if \(a|b.\)
Determine whether the following relation is a poset. Let the relation \(R\) on the set of numbers \(\{1,2,3,5,6,10,15,30\}\) be defined by \(aRb\) if and only if \(a|b.\)
Determine whether the following relation is a poset. Let the relation \(R\) on the set of positive integers be defined by \(aRb\) if and only if \(a|b.\)
Determine whether the following relation is a poset. Let the relation \(R\) on the set of real numbers be defined by \(aRb\) if and only if \(a \le b.\)
Determine whether the following relation is a poset. Consider the set \(\Pow(X)\) for the set \(X=\{a,b,c\}.\) Let the relation \(R\) be defined by \(A\) is related to \(B\) if and only if \(A \subseteq B\text{.}\)
An element \(c\) is a least upper bound of two elements \(a\) and \(b\text{,}\) denoted \(a \wedge b\text{,}\) if and only if \(a \preceq c,\)\(b \preceq c\) and if \(a \preceq d,\)\(b \preceq d\) for any other element then \(c \preceq d.\)
For the poset represented by the Hasse diagram in FigureΒ 4.4.2\(lub(a,b)=d\text{.}\) Note h is another upper bound of a and b, but \(d \preceq h\text{.}\)
An element \(c\) is a greatest lower bound of two elements \(a\) and \(b\text{,}\) denoted \(a \vee b\text{,}\) if and only if \(c \preceq a,\)\(c \preceq b\) and if \(d \preceq a,\)\(d \preceq b\) for any other element then \(d \preceq c.\)
For the poset represented by the Hasse diagram in FigureΒ 4.4.2 the minimal elements are a,b, and c. a is a minimal element because there is no element x such that \(x \preceq a\text{.}\) Notice where the minimal elements are in the Hasse diagram.
For the poset represented by the Hasse diagram in FigureΒ 4.4.2 the maximal elements are f,g, and h. f is a maximal element because there is no element x such that \(f \preceq x\text{.}\)