1.1.2. Question 1: Equivalence Relations - Part A

Try these first

1. 1.

   What properties must an equivalence relation satisfy?

   Hint

   Think about the initials 'RST'.

2. 2.

   Give an example of a reflexive relation.

   Hint

   Consider any single element of a set.

3. 3.

   What is an equivalence relation?

   • Only reflexive
   • Reflexive
   • symmetric
   • and transitive
   • None of the above

   Hint

   Recall the three properties we discussed.

4. 4.

   True or False: The union of two equivalence relations is always an equivalence relation.

   • True
   • False

   Hint

   Think of the counterexamples we discussed.

5. 5.

   Suppose R1 and R2 are two equivalence relations defined on the set {x, y, z}. Describe the union and intersection of these relations and show whether they remain equivalence relations.

   Hint

   Use specific pairs from both relations to illustrate.

6. 6.

   Provide a scenario where the union of two equivalence relations fails transitivity and prove your point with an example.

   Hint

   Identify which pairs are present and which are missing.