23.2.9. Comparable and Incomparable Elements

Try these first

1. 1.

   Define reflexivity in terms of partial ordering.

   Hint

   Think about how your name relates to you.

2. 2.

   What is the meaning of incomparable elements?

   Hint

   Consider numbers that do not divide each other.

3. 3.

   What must a relation satisfy to be a partial ordering?

   • Reflexivity only
   • Reflexivity
   • Antisymmetry
   • and Transitivity
   • Any relationship

   Hint

   Recall the acronym R-A-T.

4. 4.

   True or False: In a total order, no elements are incomparable.

   • True
   • False

   Hint

   Think about the definition of total ordering.

5. 5.

   Given the set S = {a, b, c} with the relation defined as 'a divides b, a divides c', construct a Hasse diagram and verify all properties.

   Hint

   Focus on visual arrangement of the nodes and the relationships they imply.

6. 6.

   List three scenarios where partial ordering would apply, ensuring to highlight incomparable elements.

   Hint

   Think of real-world examples that show relationships and dependencies.