22.5. Proof of Properties of Equivalence Relation

Try these first

1. 1.

   Define a partition of the set {a, b, c, d}.

   Hint

   Make sure the subsets do not overlap.

2. 2.

   What does it mean for two subsets to be pairwise disjoint?

   Hint

   Consider the definition of disjoint sets.

3. 3.

   What is an equivalence relation?

   • A relation that is only reflexive
   • A relation that is symmetric and transitive
   • A relation that is reflexive
   • symmetric
   • and transitive

   Hint

   Think about the definitions you've learned.

4. 4.

   True or False: Every partition of a set results in overlapping subsets.

   • True
   • False

   Hint

   Recall the definition of a partition.

5. 5.

   Given a set C with five distinct elements, explore how many different partitions can be formed. Provide your reasoning.

   Hint

   Consider small cases first to build understanding.

6. 6.

   Formulate an equivalence class relation from the set of natural numbers N which relays elements by their parity. Prove it is an equivalence relation.

   Hint

   Check if the definitions hold according to the properties of equivalence relations.