23.3.2. Example with Less than Equal To Relationship

Try these first

1. 1.

   What is reflexivity in partial ordering?

   Hint

   Think about definitions and how they relate to equality.

2. 2.

   Give an example of two comparable elements.

   Hint

   Consider numbers and the ordering property.

3. 3.

   What property states that each element must relate to itself?

   • Antisymmetry
   • Reflexivity
   • Transitivity

   Hint

   Think about how self-identities function.

4. 4.

   In a partial ordering, if A � B and B � C, then A � C represents which property?

   • Antisymmetry
   • Reflexivity
   • Transitivity

   Hint

   Consider the movement of relationships.

5. 5.

   Given the relationship of divisibility among integers, create a Hasse diagram for the set of integers {1, 2, 3, 6, 12}. Explain the reasoning behind the placement of each integer.

   Hint

   Think about the layers of divisibility.

6. 6.

   Explain how adding an additional element to a partially ordered set changes its structure. Can you illustrate with the divides relationship?

   Hint

   Consider how additional layers interact with existing elements.