Lecture 16 21 127 Concepts of Math 10 05 2012 Administrivia Overall we were very happy with how well you did on the exam Remember that regrade submissions are due on Monday That is a strict deadline Here are the grade cutoffs A 179 B 154 C 125 D 92 Chapter 5 is posted Read the part about relations and equivalence relations over the weekend Skip the part about order relations Relations and Properties 1 Define the relation R on a b c by R a a a c b c c b This is not reflexive not symmetric and not transitive 2 Define the relation on the set S of people in this room by saying for any x y S x y x and y were born in the same month This is reflexive symmetric and transitive 3 Define the relation on P N by saying for any X Y N X Y X Y X Y For example 1 1 2 3 and 1 2 3 1 2 3 4 5 and 1 2 3 5 6 7 but 1 2 3 6 1 2 and 4 5 6 6 5 6 7 This is reflexive not symmetric and not transitive Equivalence Relations Definition An equivalence relation is a relation that is reflexive symmetric and transitive The idea is that an equivalence relation packages the underlying set into pieces Example 0 1 on the set R define R to be the relation where x y R bxc byc Notice that this relation is 1 reflexive because x R bxc bxc symmetric because x y R if bxc byc then certainly byc bxc transitive because x y z R if bxc byc and byc bzc then certainly bxc bzc Notice that all reals are related to an integer and any two reals related to the same integer are related to each other We can package all the reals 0 x 1 together and all the reals 1 x 2 and all the reals 1 x 0 and so on We can represent each of those sets by one representative element Example 0 2 Recall the born in the same month relation This packages everyone into classes based on what month they were born in Equivalence Classes Definition Let R be an equivalence relation on the set A Let x A The equivalence class under R corresponding to x is x R y A x y R Look back at the previous examples from this lecture Here are some examples of equivalence classes 0 R y R 0 y 1 October birthmonth Brendan Definition Let R be an equivalence relation on the set A The set of equivalence classes under R denoted by A R is A modulo R That is A R x R x A Equivalently A R X A x A X x R Example 0 3 Again look at the previous examples R R 2 R 1 R 0 R 1 R 2 R T R x Z T x R Notice that x R R x Z and x y Z x 6 y x R y R note that this is much stronger than x R 6 y R This property holds with the other example too students S modulo birth month yields a set of 12 equivalence classes each represented by someone who was born in that month Again notice that the union of the classes yields the whole class everyone was born in some month and the classes are pairwise disjoint everyone was born in at most one month Definition Let A be a set A partition of A is a collection of sets that are pairwise disjoint and whose union is A That is a partition is formed by an index set I and non empty sets Si defined for every i I that satisfy i I Si A and i j I i 6 j Si Sj and Si A i I 2 Theorem Suppose R is an equivalence relation on A Then the equivalence classes the elements of A R form a partition of A Proof On your homework We will prove the converse in class This means the Theorem is conveniently an claim Theorem Suppose we have a partition of a set A Then those sets define an equivalence relation on A specifically the sets are the equivalence classes Said another way suppose we have a set A and a partition F Then there exists an equivalence relation R on A such that A R F 3
View Full Document
Unlocking...