24.4.2.1. Finding Minimal Elements

Try these first

1. 1.

   Define a poset.

   Hint

   Recall the three properties of the relation.

2. 2.

   What does it mean for an element to cover another?

   Hint

   Think about visualizing it in a Hasse diagram.

3. 3.

   What defines a partially ordered set (poset)?

   • Reflexive relation only
   • Transitive relation only
   • Reflexive
   • antisymmetric
   • and transitive relation

   Hint

   Think of the definition of a poset.

4. 4.

   True or False: Every maximal element must also be a cover.

   • True
   • False

   Hint

   Consider the definitions carefully.

5. 5.

   Create a poset using the elements {A, B, C, D} such that you have at least two covers for one of the elements. Identify the covers and explain your structure.

   Hint

   Consider how you can relate elements while keeping some independent.

6. 6.

   In a poset where X < Y and X < Z, if Y and Z are not related, identify the maximal elements. Then, determine if there can be a unique least element in this poset.

   Hint

   Visualize a scenario where you can see independent relationships.