Lecture 17 21 127 Concepts of Math 10 08 2012 Administrivia HW 5 due Thursday Lecture recitation notes from last week are posted Equivalence Relations Define on R R by x y u v x u y v Show its properties Draw a picture to show the points related to a given x y Note that those points aren t all related to each other Thus it is not an equivalence relation This is good intuition and scratch work Define on R R by x y u v x u Show its properties It is an equivalence relation Draw a similar picture and notice that all of the points related to a given x y are indeed related to each other Every point on the horizontal axis has a corresponding set of points an equivalence class Write draw some examples and characterize R R One last example Define on R R by x y u v p p x2 y 2 u2 v 2 p This is the has the same distance to the origin relation In mathematics we call that function x2 y 2 a metric This is also an equivalence relation and every real z 0 has a corresponding equivalence class so we use R as the representatives Draw some equivalence classes point out how two classes are the same or else disjoint and every class contains itself This is what you are doing on homework Theorem and Proof with Example Definition A partition of a set S is a collection of sets Si for every i I where I is some index set that satisfy i I Si S and Si 6 the sets are non empty subsets of S and i j I i 6 j Si Sj the sets are pair wise disjoint and Si S i I the sets cover S Point out how pair wise disjoint is very different from having an empty interesection 1 Example 0 1 Consider the set S 6 Define the collection of sets F 1 4 2 3 5 6 Notice that F is a partition of S because the sets are disjoint none are empty and their union is S Wouldn t it be nice if there were some equivalence relation R that yielded these sets when we considered S R It turns out that there is Of course we may not be able to define it in a nice way like the relations we have seen so far which are usually defined as x y R x and y share some common property However having the partition in hand already allows us to define the relation in terms of the partition Specifically the partition sets are the equivalence classes The partition itself builds in the equivalence class structure and we can just define an equivalence relation R by saying x y R x and y belong to the same partition set In this example we would define S1 1 4 and S2 2 3 5 and S3 6 Then we define the relation R but x y R i 3 x Si y Si Now we re ready to state and prove a theorem Theorem 0 2 Let S be a set and let F be a partition of S Then there exists an equivalence relation R such that S R F Proof Let F be a partition of S This means we have an index set I and F Si i I where the sets Si satisfy Si S and Si 6 and Si S and i j I i 6 j Si Sj i I Let s define the relation R on S by x y R i I x Si y Si We will now prove R is an equivalence relation Let x S be arbitrary and fixed Since the sets Si cover S we know i I x Si Certainly then x Si and x Si so x x R Therefore R is reflexive Let x y S be arbitrary and fixed Suppose x y R This means i I x Si y Si Certainly then for that i I we have y Si x Si Thus y x R as well Therfore R is symmetric Let x y z S be arbitrary and fixed Suppose that x y R and y z R This means i I x Si y Si and j I y Sj z Sj Notice that y Si y Sj Since Si Sj for any distinct i j it must be that i j Otherwise y which is impossible Accordingly x Si and y Si and z Si Thus x z R Therefore R is transitive Since all three properties hold R is an equivalence relation The equivalence classes of S modulo R S R are of the form x R where x S Since F is a partition of S x Si for some i Thus x R Si for some i Therefore all the equivalence classes are equal to some set Si Likewise any set Si 6 so x Si and thus there is a corresponding equivalence class Si x R Therefore every equivalence class is a set of the form Si and vice versa This shows that any partition corresponds nicely to an equivalence relation and its classes On homework you will prove the converse That is you will prove that when we have an equivalence relation the equivalence classes partition the overall set 2
View Full Document
Unlocking...