23.2.8. Abstract Notation for Relations

Try these first

1. 1.

   What is reflexivity in relation to partial ordering?

   Hint

   Think about how many checks and balances exist in defining relationships.

2. 2.

   Give an example of transitivity.

   Hint

   Think of a lineup where people cannot jump ahead.

3. 3.

   What is the definition of partial ordering?

   • A strict linear order
   • A relation with reflexivity
   • antisymmetry
   • and transitivity
   • Any random set

   Hint

   Consider the properties that define partial relations.

4. 4.

   Are the elements 2 and 3 comparable in partial order of divisibility?

   • True
   • False

   Hint

   Think about the relationships that can exist.

5. 5.

   Define a new relation on the set of integers where A ≤ B means A can be grouped to form B. Prove that this relation is a partial order.

   Hint

   Think about how fractions can form larger sets for grouping.

6. 6.

   Given the subset relation, identify the Hasse diagram for {1, 2, {1, 2}}. Clarify all relationships.

   Hint

   Reflect on how smaller sets fit within larger ones, like visualizing building blocks.