22.2. Introduction

Try these first

1. 1.

   Define an equivalence relation. List its three properties.

   Hint

   Think of properties that a relation must hold for elements to be considered equivalent.

2. 2.

   What is a partition? Give a real-world example.

   Hint

   Consider how objects can be grouped without overlaps.

3. 3.

   What are the three properties of an equivalence relation?

   • Reflexivity
   • Disjointness
   • Transitivity
   • Reflexivity
   • Symmetry
   • Transitivity
   • Symmetry
   • Pairwise
   • Union

   Hint

   Remember the definitions of each property.

4. 4.

   True or False: A partition can have overlapping subsets.

   • True
   • False

   Hint

   Think about how subsets in a partition interact with each other.

5. 5.

   Take the set {3, 5, 7, 9} and define a relation that is an equivalence relation. Prove it is reflexive, symmetric, and transitive.

   Hint

   Focus on the definitions of the properties and how they apply to odd numbers.

6. 6.

   Given a set of animals [Dog, Cat, Cow] and their classification [Mammal, Reptile], form partitions based on their classifications. Explain the equivalence relation established.

   Hint

   Think about how animals relate to classifications and how this forms an equivalence.