A subset \(K\) of \(J\) is said to be cofinal in \(J\) if for each \(\alpha \in J\), there exists \(\beta \in K\) such that \(\alpha \preceq \beta\). Show that if \(J\) is a directed set and \(K\) is cofinal in \(J\), then \(K\) is a directed set.
Solution
1. Definition and properties of a directed set:
1.1. A set \((J, \preceq)\) is a directed set if:(i) \(\preceq\) is a preorder on \(J\) (reflexive: \(\alpha \preceq \alpha\), and transitive: \(\alpha \preceq \beta \land \beta \preceq \gamma \implies \alpha \preceq \gamma\)), and(ii) for every pair \(\alpha_1, \alpha_2 \in J\), there exists an upper bound \(\gamma \in J\) such that \(\alpha_1 \preceq \gamma\) and \(\alpha_2 \preceq \gamma\).
definition of a directed set.
1.2. The subset \(K \subseteq J\) inherits the relation \(\preceq\).Since reflexivity and transitivity hold on all elements of \(J\), they hold on all elements of \(K\).
restriction of a preorder to a subset.
2. Existence of common upper bounds in \(K\):
2.1. Let \(k_1, k_2 \in K\) be arbitrary elements.Since \(K \subseteq J\), \(k_1, k_2 \in J\).
subset containment.
2.2. Since \(J\) is directed, there exists an element \(\alpha \in J\) such that:\[ k_1 \preceq \alpha \quad \text{and} \quad k_2 \preceq \alpha. \]
directedness of \(J\).
2.3. Since \(K\) is cofinal in \(J\), there exists an element \(\beta \in K\) such that:\[ \alpha \preceq \beta. \]
definition of cofinality of \(K\) in \(J\).
2.4. By transitivity of \(\preceq\):\[ k_1 \preceq \alpha \text{ and } \alpha \preceq \beta \implies k_1 \preceq \beta, \]\[ k_2 \preceq \alpha \text{ and } \alpha \preceq \beta \implies k_2 \preceq \beta. \]
transitivity of preorder \(\preceq\).
2.5. Thus \(\beta \in K\) is a common upper bound for \(k_1\) and \(k_2\) in \(K\).
3 and 4.
3. Conclusion:\(K\) with the inherited relation is a directed set. Q.E.D.
1 and 2.