24.4.3. Proof of Compatibility with Original Relation

Try these first

1. 1.

   Define what a cover is in the context of posets.

   Hint

   Think about elements immediately above others in a diagram.

2. 2.

   What is a maximal element?

   Hint

   Consider elements without anything on top in a Hasse diagram.

3. 3.

   What defines a cover in a poset?

   • x is related to y
   • x is maximal
   • x is minimal

   Hint

   Refer to the definitions we've just discussed.

4. 4.

   True or False: A poset can have more than one minimal element.

   • True
   • False

   Hint

   Consider how many elements can have no dependency below them.

5. 5.

   Design a Hasse diagram for a given set E = {a, b, c} with relations a < b, a < c, and b < c. Identify the covers, maximal, and minimal elements.

   Hint

   Focus on the relationship levels.

6. 6.

   Given a set of tasks with dependencies described, explain how a circular dependency impacts the topological sort and provide a solution.

   Hint

   What happens if tasks rely on one another in a loop?