21.4. Equivalence Classes

Try these first

1. 1.

   What does it mean for a relation to be reflexive?

   Hint

   Think about how elements relate to themselves.

2. 2.

   List one property of equivalence relations.

   Hint

   Recall the three key properties.

3. 3.

   Which property is NOT required for a relation to be an equivalence relation?

   • Transitivity
   • Symmetry
   • Asymmetry

   Hint

   Think about the definitions of these properties.

4. 4.

   True or False: Two equivalence classes can share some elements.

   • True
   • False

   Hint

   Recall the property of equivalence classes regarding their separation.

5. 5.

   Given the set of integers Z and the relation defined by modulo 5, calculate and list all equivalence classes.

   Hint

   Consider what numbers share the same remainder when divided by 5.

6. 6.

   Prove or disprove: There exists integers a and b such that a ≡ b (mod m) but a is not equal to b.

   Hint

   Look at how properties of congruence work under different moduli.