21.6. Disjoint Equivalence Classes

Try these first

1. 1.

   What are the three properties of an equivalence relation?

   Hint

   Think about how elements relate to themselves and each other.

2. 2.

   What does it mean for two sets to be disjoint?

   Hint

   Consider how they can exist with no overlap.

3. 3.

   What property guarantees that every element is related to itself?

   • Symmetry
   • Reflexivity
   • Transitivity

   Hint

   Think of the 'self' aspect of a relation.

4. 4.

   True or False: Equivalence classes can share some common elements.

   • True
   • False

   Hint

   Consider the definitions of equivalence classes.

5. 5.

   Prove that under any equivalence relation, the union of equivalence classes covers the entire set A.

   Hint

   Consider every element's relation to itself.

6. 6.

   Given an equivalence relation defined on a set of integers, show that any integer belongs to only one equivalence class.

   Hint

   Examine how equivalence classes link back to remainders.