22.5.3. Transitivity

Try these first

1. 1.

   Define a partition of the set {1, 2, 3, 4}.

   Hint

   Ensure subsets do not overlap.

2. 2.

   What does reflexivity mean in the context of equivalence relations?

   Hint

   Think of how this applies to any set.

3. 3.

   What is a partition of a set?

   • A group of overlapping subsets
   • Non-empty disjoint subsets whose union is the set
   • Only one subset of the whole set

   Hint

   Consider the properties of subsets in your definition.

4. 4.

   True or False: Every equivalence relation's equivalence classes can form a partition.

   • True
   • False

   Hint

   Recall what defines a valid partition.

5. 5.

   Given the set {x, y, z, a, b}, define a partition and illustrate its equivalence classes.

   Hint

   Be sure to show how elements relate within these classes demonstrating non-overlapping subsets.

6. 6.

   Construct an equivalence relation from the partition {{1, 3}, {2, 4}} and prove it is reflexive, symmetric, and transitive.

   Hint

   Carefully outline each property while validating your relationships.