24.5.1. Partial Ordering

Try these first

1. 1.

   Define what a poset is.

   Hint

   Remember the acronym RAT.

2. 2.

   What does it mean for one element to cover another in a poset?

   Hint

   Think diagrammatically about their relationship.

3. 3.

   What three properties must a relation possess to be a partial order?

   • Reflexive
   • Symmetric
   • Transitive
   • Reflexive
   • Antisymmetric
   • Transitive
   • Symmetric
   • Asymmetric
   • Transitive

   Hint

   Remember the RAT acronym to help.

4. 4.

   True or False? A minimal element can also be a maximal element in a poset.

   • True
   • False

   Hint

   Visualize it in a Hasse diagram.

5. 5.

   Construct a Hasse diagram for the set of integers {1, 2, 3, 4, 5} under the relation defined by divisibility.

   Hint

   Consider divisor relationships to create this diagram.

6. 6.

   If a poset has maximal elements 6 and 7 with no element covering 5, what can you deduce about the relationship between these elements?

   Hint

   Analyze cover relationships to conclude independencies.