91Ó°ÊÓ

Determine whether the indicated sets and relations give examples of partially ordered sets. The set of integers \(\mathbb{Z}\) with \(a \leq b\) to mean \(a\) divides \(b\).

Determine whether the indicated sets and relations give examples of partially ordered sets. The set of natural numbers \(\mathbb{N}\) with \(a \leq b\) to mean \(a\) divides \(b\).

Determine whether the indicated sets and relations give examples of partially ordered sets. The set of natural numbers \(\mathbb{N}\) with \(a \leq b\) to mean \(a\) divides \(b\).

Determine whether the indicated set with the indicated relation is a lattice. The set of all positive divisors of 70 with \(a \leq b\) to mean \(a\) divides \(b\).

Determine whether the indicated set with the indicated relation is a lattice. The set of all positive divisors of 60 with \(a \leq b\) to mean \(a\) divides \(b\).

Determine whether the indicated set with the indicated relation is a lattice. The set \(L \times M=\\{(a, b) \mid a \in L, b \in M\\},\) where \(L\) and \(M\) are lattices, with \((a, b) \leq(c, d)\) to mean that \(a \leq c\) in \(L\) and \(b \leq d\) in \(M\).

Show that a lattice \(L\) is distributive if and only if for all \(a, b, c \in L\) (Cancellation) \(a \wedge b=a \wedge c\) and \(a \vee b=a \vee c\) implies \(b=c\)

Construct the Hasse diagram for the lattice of subgroups of \(S_{3}\)

Construct the Hasse diagram for the lattice of subgroups of \(D_{4}\)

Draw the Hasse diagrams of all nonisomorphic Boolean algebras of orders \(|B|=2,4,\) or 8.