1.1.4. Question 2: Union and Composition of Equivalence Relations

Try these first

1. 1.

   Define an equivalence relation.

   Hint

   Think about each property individually.

2. 2.

   What is the result of the union of two equivalence relations?

   Hint

   Consider how pairs combine.

3. 3.

   Is the union of two equivalence relations always an equivalence relation?

   • True
   • False

   Hint

   Think of specific pairs that could violate this.

4. 4.

   What property must hold for R1 ∪ R2 to be an equivalence relation?

   • It must be reflexive
   • It must be symmetric
   • R1 ∘ R2 = R1 ∪ R2
   • Both A and B

   Hint

   Consider how both relations interact.

5. 5.

   Prove the statement: If R1 is a subset of R2 and both are equivalence relations, what can you say about R1 ∪ R2 and R2?

   Hint

   Consider subset relationships.

6. 6.

   Construct two equivalence relations R1 and R2, where the union is not transitive. Provide elements that demonstrate this.

   Hint

   Identify ordered pairs that fail to connect transitive links.