21.5. Examples of Equivalence Classes

Try these first

1. 1.

   What does it mean for a relation to be reflexive?

   Hint

   Think about a number interacting with itself.

2. 2.

   Give an example of an equivalence class in mod 3.

   Hint

   Consider what results when you divide the integers by 3.

3. 3.

   What property states that if a is related to b, then b is also related to a?

   • Reflexivity
   • Symmetry
   • Transitivity

   Hint

   Think about how relationships mirror each other.

4. 4.

   True or False: An equivalence class can be empty.

   • True
   • False

   Hint

   Recall the property of every equivalence class.

5. 5.

   Given the relation a ≡ b (mod 4), determine whether pairs (6, 10) and (2, 6) belong to the same equivalence class. Justify your answer.

   Hint

   Check the remainder of each number when divided by 4.

6. 6.

   Prove that all integers can be classified under an equivalence relation based on their remainders when divided by n for any integer n > 0.

   Hint

   Utilize a proof by contradiction regarding shared remainders.