22.4.1. Equivalence Classes as Partitions

Try these first

1. 1.

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

   Hint

   Think about the groups that can be formed without overlap.

2. 2.

   List two properties of a partition.

   Hint

   Recall the definition of a partition.

3. 3.

   What is a partition?

   • A collection of overlapping subsets
   • A collection of non-empty disjoint subsets
   • A single set only

   Hint

   Remember the definition of a partition.

4. 4.

   Do equivalence classes overlap?

   • True
   • False

   Hint

   Think about the properties of sets in a partition.

5. 5.

   Given set D = {1, 2, 3, 4, 5}, partition D into four subsets. Construct an equivalence relation from this partition.

   Hint

   For each subset in the partition, create the relations as described in the lesson.

6. 6.

   Imagine a relation R partitions a set E into {A, B, C, D}. Provide examples of what the subsets might look like and prove they form equivalence classes.

   Hint

   Draw diagrams to visualize the partitions and equivalence classes.