2.3. Question 7: Equivalence Relation

Try these first

1. 1.

   Define an equivalence relation.

   Hint

   Think about the three properties that must be satisfied.

2. 2.

   What does it mean to partition a set?

   Hint

   Consider how groups of elements can be formed.

3. 3.

   What defines an equivalence relation?

   • Only reflexive
   • Reflexive
   • symmetric
   • and transitive
   • Only symmetric

   Hint

   Remember the properties we discussed using RST.

4. 4.

   Does every partition of a set form an equivalence relation?

   • True
   • False

   Hint

   Think about how each element is categorized.

5. 5.

   Given a set with 50 elements, partition this set into four equal subsets and find the number of ordered pairs within each subset.

   Hint

   Break down the problem by first finding the size of each subset.

6. 6.

   Prove that if a relation R on a set A is reflexive and symmetric, it does not imply R is transitive. Provide a counterexample.

   Hint

   Reflect on how pairing can lead to violations of transitivity.