1.4. Hasse Diagrams and their Categories

Try these first

1. 1.

   Describe a Hasse diagram with three elements that shows no relationships.

   Hint

   Think of three separate dots.

2. 2.

   What makes a relation a partial order?

   Hint

   Consider the definitions of these properties.

3. 3.

   What is a Hasse diagram?

   • A type of graph for functions
   • A way to visualize partially ordered sets
   • An equation

   Hint

   Think of how we visualize elements and their order.

4. 4.

   A relation must be reflexive, antisymmetric, and transitive to be considered a?

   • Total order
   • Partial order
   • Linear chain

   Hint

   Look at the definitions of the properties.

5. 5.

   Construct Hasse diagrams for the set {x, y, z} where x is the least element. Identify the relationships.

   Hint

   Visualize how many nodes are connected and their direct order.

6. 6.

   Prove that a set with n elements can have a maximum of 2^n - 1 distinct Hasse diagrams.

   Hint

   Consider how many ways you can compare each pair of elements.