1.1.1. Introduction to Tutorial 4: Part I

Try these first

1. 1.

   List the three properties of equivalence relations.

   Hint

   Think about what it means for a relation to be an equivalence.

2. 2.

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

   Hint

   Consider if the union could ever lose transitivity.

3. 3.

   What are the three properties required for a relation to be an equivalence relation?

   • Reflexivity
   • Symmetry
   • Transitivity
   • Completeness
   • Reflexivity
   • Antisymmetry

   Hint

   Think about the definitions we've discussed.

4. 4.

   True or False? The intersection of two equivalence relations is always an equivalence relation.

   • True
   • False

   Hint

   Consider how the properties overlap.

5. 5.

   Define a custom set and create two distinct equivalence relations on it. Analyze and demonstrate their union and intersection.

   Hint

   Work through constructing the relations step by step.

6. 6.

   Prove that the union of two equivalence relations forms a new relation that does not alter the existing reflexive pairs.

   Hint

   Check the properties of union through individual reflexive elements.