22.4.2.1. Construction of the Equivalence Relation

Try these first

1. 1.

   What is a partition of a set?

   Hint

   Think of how a pizza can be cut into slices.

2. 2.

   Does a partition of a set have overlapping subsets?

   Hint

   Consider how two different states cannot overlap.

3. 3.

   What is an equivalence relation?

   • A relation that is only symmetric.
   • A relation that is reflexive
   • symmetric
   • and transitive.
   • A relation that does not require any properties.

   Hint

   Recall the properties we discussed.

4. 4.

   True or False: A partition can have overlapping subsets.

   • True
   • False

   Hint

   Think of how subsets cannot share elements.

5. 5.

   Construct a partition of the set {1, 2, 3, 4, 5} into subsets of size two.

   Hint

   Think about how you can pair up elements.

6. 6.

   Prove that the equivalence classes of an equivalence relation form a partition.

   Hint

   Use the properties of equivalence relations in your proof.