22.5.1. Reflexivity

Try these first

1. 1.

   What is a partition of a set?

   Hint

   Think about how subsets can combine back to the original set.

2. 2.

   Define an equivalence relation.

   Hint

   Consider the properties of relationships.

3. 3.

   What is a characteristic of a partition?

   • They must overlap.
   • They are non-empty.
   • They can contain empty sets.

   Hint

   Remember the requirements for a partition.

4. 4.

   True or False: An equivalence relation requires that all elements relate to each other.

   • True
   • False

   Hint

   Consider how equivalence classes function.

5. 5.

   Given a set A = {1, 2, 3, 4, 5}, create all possible partitions and identify their equivalence classes.

   Hint

   Start by generating smaller partitions and build from there.

6. 6.

   Prove that if you have a partition of a set, you can produce an equivalence relation whose equivalence classes are those subsets.

   Hint

   Use examples from known partitions to visualize your explanation.