23.2.10. Definition of Total Ordering

Try these first

1. 1.

   Define total ordering in your own words.

   Hint

   Think about relationships in sets.

2. 2.

   What property states that A ≤ A?

   Hint

   It's about self-relation.

3. 3.

   Which of the following is an example of total ordering?

   • The subset relation
   • Less than or equal relation
   • Divides relation

   Hint

   Consider which relationships have definite comparisons.

4. 4.

   True or False: Total ordering can have incomparable elements.

   • True
   • False

   Hint

   Think about what totality means in terms of element comparison.

5. 5.

   Consider the set {1, 2, 3, 4}. Identify and justify which ordering relations classify it as a total order.

   Hint

   Write out comparisons for pairs and check connectivity.

6. 6.

   Find a set where divisibility does not provide total ordering. Explain the elements and relationships.

   Hint

   Think of how some numbers do not relate under the divides relation.