21.3. Properties of Equivalence Relations

Try these first

1. 1.

   What are the three properties of an equivalence relation?

   Hint

   Use the acronym RST.

2. 2.

   Is the relation R defined as (a, b) such that a = b an equivalence relation?

   Hint

   Consider if each property holds true.

3. 3.

   Which of the following is NOT a property of equivalence relations?

   • Reflexive
   • Irreflexive
   • Symmetric
   • Transitive

   Hint

   Recall the properties we discussed.

4. 4.

   True or False: An equivalence class can be empty.

   • True
   • False

   Hint

   Think about how classes are formed based on relations.

5. 5.

   Show that the relation R on the set of all people where a R b if they are the same age is an equivalence relation.

   Hint

   Check each property carefully.

6. 6.

   Given the equivalence relation defined by modular arithmetic, form and describe equivalence classes for five distinct integers under modulo 5.

   Hint

   Picture how numbers relate to the modulus.