Lecture 20: EquivalencesIn this lecture, we will revise some of the concepts on relations that we covered previously.
Then we will look into equivalence relations and equivalence classes. We will see how an equivalence on a set partitions the set into equivalence classes. RevisionLet us make sure we understand key concepts before we move on. To do so, take five minutes to solve the following problems on your own. You are welcome to discuss your solutions with me after class.
Equivalence RelationAn equivalence relation over a set is a reflexive, symmetric and transitive relation. Question
Which of the following are examples of equivalence relations over .
Just check that the relations above are reflexive, symmetric and transitive. Answer
Example1The relation is an equivalence relation. The Cartesian product of any set with itself is a relation . All possible tuples exist in . This relation is also an equivalence. ModularCongruencesFor any number , we have an equivalence relation . Often we denote by the notation (read as and are congruent modulo ). Verify that is an equivalence for any . Reflexivity: For all , we have that . Therefore . Symmetry: Clearly if then . Transitivity: If and then . Equivalence ClassesWe now look at how equivalence relation on partitions the original set . Let us take the set . And the equivalence . Let us collect everything that is equivalent to. This gives us the set . Collecting everything equivalent to again gives us . Collecting everything that is equivalent to gives us and similarly for , we get . Therefore, is said to induce the following partition of the set : Question
What is the partition induced by the equivalence: ? To answer this, consider:
The partition induced by is therefore: . Notation
The equivalence class of an element under an equivalence relation is denoted as . Claim1
If then . Proof
Let . We show that and vice versa, . To show that , let . We have . Since , we conclude by transitivity of that . Therefore . The proof of is very similar. Therefore represent the same equivalence classes. We now show that two equivalence classes are either the same or disjoint. Claim2
Whenever then . {Proof} Let . We have and . Therefore by symmetry and transitivity of , we conclude that . Therefore . This shows that given any set , and equivalence relation , the equivalence classes is a partition of the set :
