1.2. Counterexamples and Properties of Equivalence Relations

Try these first

1. 1.

   Define reflexivity in the context of equivalence relations.

   Hint

   Think of how each element compares to itself.

2. 2.

   Is the union of two equivalence relations always reflexive? Why or why not?

   Hint

   Consider what reflexivity requires.

3. 3.

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

   • Reflexivity
   • symmetry
   • transitivity
   • Reflexivity
   • antisymmetry
   • symmetry
   • Symmetry
   • transitivity
   • antisymmetry

   Hint

   Recall the definitions of each property.

4. 4.

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

   • True
   • False

   Hint

   Think about counterexamples demonstrating transitivity issues.

5. 5.

   Define equivalence relations and provide a real-world scenario where such relations apply. Then analyze if the union of the scenarios forms an equivalence relation.

   Hint

   Explore connections through examples relevant to education or social classes.

6. 6.

   Create a recursive structure to count equivalence relations on a set of 4 elements, P(4), using the provided function P(n).

   Hint

   Break down into selected subsets and analyze how remaining elements are partitioned.