24.1.2. Properties of Covers

Try these first

1. 1.

   What is a poset?

   Hint

   Think about the properties that define relationships.

2. 2.

   What does it mean when we say 'y covers x'?

   Hint

   Visualize in terms of layers.

3. 3.

   What defines a cover in a poset?

   • A relationship with a middle element
   • Immediate upward relation with no intermediaries
   • Directly connected elements only

   Hint

   Think about Hasse diagrams.

4. 4.

   True or False: A poset can have multiple maximal elements.

   • True
   • False

   Hint

   Recall the definition of maximal.

5. 5.

   Construct a simple Hasse diagram for a poset containing the elements {1, 2, 3, 4} with the following relationships: 1 < 2, 1 < 3, 2 < 4, 3 < 4. Identify maximal and minimal elements.

   Hint

   Draw it layer by layer and look for the top and bottom.

6. 6.

   Given the elements {a, b, c, d} where a < b, a < c, b < d, but c is not related to d, demonstrate whether d can be a greatest element.

   Hint

   Check the relationships to determine coverage.