21. Equivalence Relation

Try these first

1. 1.

   What are the three properties that define an equivalence relation?

   Hint

   Think of the acronym RST.

2. 2.

   Provide an example of an equivalence relation on the set of integers.

   Hint

   Consider how integers can relate based on remainders.

3. 3.

   What property states that every element is related to itself?

   • Reflexive
   • Symmetric
   • Transitive

   Hint

   It's one of the RST properties.

4. 4.

   True or False: All elements in an equivalence class must be related to at least one other element.

   • True
   • False

   Hint

   Consider the definition of an equivalence class.

5. 5.

   Prove that the relation defined as (a, b) if a - b is divisible by 7 is an equivalence relation.

   Hint

   Assume a, b, c are any integers and verify each property.

6. 6.

   Consider a set of students where two students are equivalent if they are in the same year. Explain whether this defines an equivalence relation.

   Hint

   Sketch out each property to clarify the relationships.