24.2.3. Existence of Maximal and Minimal Elements

Try these first

1. 1.

   Define a maximal element in a poset with an example.

   Hint

   Think about the largest number in a small set.

2. 2.

   What is a minimal element?

   Hint

   Consider the smallest number in a sorted list.

3. 3.

   What is a maximal element in a poset?

   • An element that is less than others
   • An element that cannot have elements above it
   • An element with no relations

   Hint

   Think of the highest point in a hierarchy.

4. 4.

   In a poset, can an element be both maximal and minimal?

   • True
   • False

   Hint

   Think about a single point set.

5. 5.

   Consider a poset defined by the relationships 1 < 2, 2 < 3, and 1 < 3. Define which elements are maximal, minimal, greatest, and least.

   Hint

   Draw a Hasse diagram to visualize.

6. 6.

   Create an example of a poset that has more than one maximal element but no least element.

   Hint

   Think about relations which do not connect all elements.