22.4. Relationship between Equivalence Relations and Partitions

Try these first

1. 1.

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

   Hint

   Remember that a partition includes disjoint sets covering the entire set.

2. 2.

   What are the properties of an equivalence relation?

   Hint

   Use the acronym RST as a memory aid.

3. 3.

   What is a key property of a partition?

   • All subsets must overlap.
   • Each subset must be non-empty and disjoint.
   • Subsets can be empty.

   Hint

   Recall the definition of a partition we discussed.

4. 4.

   If two equivalence classes intersect, what can we conclude?

   • True
   • False

   Hint

   Think about the definition of equivalence classes.

5. 5.

   Prove that if the equivalence relation R has classes of size 1, then R is trivial.

   Hint

   Consider what it means for elements to be only related to themselves.

6. 6.

   For the set {1, 2, 3, 4, 5}, can you create at least three distinct partitions and describe the equivalence relation for each?

   Hint

   Focus on how elements belong to each class based on your subsets.