Partially Ordered Sets · Partial Orders
Lesson 8
A chain is a subset of a partially ordered set, any two elements of which are comparable, and an antichain is a subset, any two elements of which are incomparable.
Problem. Consider the following poset \((X, \preceq)\): \(X=2^{\{a, b, c\}}\) (this is the set of all subsets of the set \(\{a, b, c\}\)) and \(x \preceq y\), if \(x \subseteq y\). Mark all true statements.
\(\{\varnothing, \{a, b\}\}\) is a chain
\(\{\{a\}, \{b\}, \{c\}\}\) is an antichain
\(\{\varnothing, \{a\}, \{b\}, \{a, b\}\}\) is a chain
\(\{\varnothing, \{a\}, \{a, b\}, \{a, b, c\}\}\) is a chain
\(\{\varnothing, \{a\}, \{b\}\}\) is an antichain
\(\{\{a, b\}, \{a, c\}, \{b, c\}\}\) is an antichain
\(\{\{a\}, \{b, c\}\}\) is an antichain