22.5.2. Symmetry

Try these first

1. 1.

   What is an equivalence relation?

   Hint

   Think about the properties it needs to satisfy.

2. 2.

   Provide a simple example of a partition.

   Hint

   Make sure the subsets do not share any elements.

3. 3.

   What is the primary property that defines an equivalence relation?

   • Only reflexivity
   • All three properties: Reflexivity
   • Symmetry
   • Transitivity
   • Only symmetry

   Hint

   Consider what each property entails.

4. 4.

   True or False: A partition can include empty subsets.

   • True
   • False

   Hint

   Think about the definition of a partition.

5. 5.

   If set A = {1, 2, 3} and set B = {3, 4, 5}, create an equivalence relation that relates elements of A to elements of B.

   Hint

   Consider how you might categorize elements across the two sets.

6. 6.

   Given a set S = {x, y, z, w} and the partition {{x, y}, {z, w}}, formulate the equivalence relation and justify its reflexivity, symmetry, and transitivity.

   Hint

   Revisit how to articulate properties based on pairs.