21.8. Theorem on Equivalence Classes

Try these first

1. 1.

   What are the three properties of an equivalence relation?

   Hint

   Think of the acronym RST.

2. 2.

   Is (2, 2) in a relation R considered reflexive?

   Hint

   Check if it holds true for any a.

3. 3.

   What property requires that an element be related to itself in an equivalence relation?

   • Symmetric
   • Transitive
   • Reflexive

   Hint

   Think of 'self-relationship'.

4. 4.

   True or False: In an equivalence relation, if a is related to b, then b must be related to a.

   • True
   • False

   Hint

   Remember the 'back and forth' characteristic.

5. 5.

   Given a set of integers and a relation defined as a ≡ b (mod 5), classify the integers based on their equivalence classes.

   Hint

   Consider the remainders when each integer is divided by 5.

6. 6.

   Prove that if two equivalence classes have any element in common, they are the same class.

   Hint

   Think about how elements are classified in equivalence relations.