1.3. Functions and Partitions

Try these first

1. 1.

   Define an equivalence relation.

   Hint

   What are the three key properties?

2. 2.

   Is the union of two reflexive relations always reflexive?

   Hint

   What does reflexivity require?

3. 3.

   What property does the union of two equivalence relations always maintain?

   • Reflexivity
   • Transitivity
   • Antisymmetry

   Hint

   Consider what it means for a relation to be reflexive.

4. 4.

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

   • True
   • False

   Hint

   Review the conditions of equivalence relations.

5. 5.

   Create an example of two equivalence relations where their union is not an equivalence relation, and specify the failure of transitivity.

   Hint

   Remember to check the interconnectedness of the elements when forming unions.

6. 6.

   Prove using contradiction that if the union of two equivalence relations isn't symmetric, then at least one of them isn't an equivalence relation.

   Hint

   Focus on how each relation interconnects and what requires them to succeed.