21.1. Definition of Equivalence Relation

Try these first

1. 1.

   Define equivalence relation.

   Hint

   What are the three key properties?

2. 2.

   Give an example of reflexivity.

   Hint

   Think of an element relating to itself.

3. 3.

   What does reflexivity imply about the elements in a set?

   • Each element is related to itself
   • All elements are related to one another
   • Elements relate in pairs

   Hint

   Think about self-relation.

4. 4.

   If a is related to b and b is related to c, what can we say about a and c?

   • True
   • False

   Hint

   Follow the chain of relations.

5. 5.

   Prove that the relation R defined on integers by a R b if a - b is divisible by 5 is an equivalence relation.

   Hint

   Use specific integers to illustrate your proof.

6. 6.

   Given two equivalence classes [a] and [b], prove that if they intersect, they must be the same.

   Hint

   Track the connections between shared elements.