Equivalence Relations ¡ Definition: A relation on a set is called an equivalence relation if it is reflexive, symmetric, and transitive. Recall the definitions: ¢ ¡ ¤ ¤ ¨ ¦ reflexive: £¥¤ §©¨ for all . ¢ ¡ ¤ ¤ ¨ ¨ ¦ ¦ ¦ £ § symmetric: £¥¤ §©¨ when , for . ¢ ¤ ¤ ¨ ¨ ¦ ¦ ¦ £ § transitive: £ §©¨ and £ § implies , ¡ ¤ ¨ for ¦ ¦ . If two elements are related by an equivalence relation, they are said to be equivalent. 1 Examples 1. Let be the relation on the set of English words such that if and only if starts with the same letter as . Then is an equivalence relation. 2. Let be the relation on the set of all human beings such that if and only if was born in the same country as . Then is an equivalence relation. 3. Let be the relation on the set of all human beings such that if and only if owns the same color car as . Then is an not equivalence relation. 2 ¤ Pr is Let ¢ ¢ an oof: is If Since Thus, equi ¤ symmetric. is By refle v ¤ alence definition, be , Congruence for xi ¤ a ¤ v positi e. some relation. £ , ¤ v then e inte £ ¤ § inte ¦ ¤ ger , § , and ger we ¤ 3 . we ha Then Modulo Using v e ha if that v the and , e this,we for ¤ relation only some ¤ ¤ proceed: if inte ger , , so and . Therefore, ¢ ¤ and If ¤ ¤ we congruence ha v e £ ¤ ¤ and , and §"! modulo £ 4 , § and , for is an inte ! is equi , then gers transiti v alence we and v £ ha e. ! v relation e . Thus, § ¦ ¡ Definition: Let be an equivalence relation on a set . The equivalence class of¤ is ¤ ¤ ¨ % ¦ £ § # $ '& ¤ % In words, # $ is the set of all elements that are related to the ¡ ¤ element ¨ . If the relation is clear, we can omit the ¤ ¤ % # $ subscript (i.e. # $ instead of ). ¤ ¨ % If # $ , then is called a representative of the equivalence class. 5 Examples Continued 1. The equivalence class of Xenon is all words starting with the letter X. That is, is an English word starting with the letter X # Xenon $ 2. The equivalence class of Chuck Cusack is all people born in the United States of America. That is, ¡ ¡ is a person that was born in the U.S.A. $ # Chuck Cusack 6 ¤ Thus, The # $ ( . congruence Example: #*2 $43 # #*) $ 0 $ + class # Congruence $43 & '& & of & & & an ¦ & ¦ & & inte , 1 ¦ ¦ 7 ¦ ger . , - Classes ¦ ¦ ¤ ) ¦ ) ¦/. ¦/. modulo ¦ ¦ ¦ ) ¦ 2 1 Modulo & ¦ & ¦/5 ¦ & & & & & ¦ & & is denoted by Equivalence Classes and Partitions ¡ Theorem 1: Let be an equivalence relation on a set . The following statements are equivalent: & - & & ¤ ¤ ¤ ) 9 8 # $ # $ # $76 # $ - - ) ) : : Proof: Show that : , , and . Notice that this theorem says that if the intersection of two equivalence classes is not empty, then they are equal. That is, two equivalence classes are either equal or disjoint. 8 of ( Definition: nonempty > is ; ¡ ¡ A is an < < < 8 6 B C a inde collections ¡ ¡ 9 subsets < ? A for x set. partition ; & ¨ = 9 ¦ F of when or > of , ; e subsets whose xample, of = 8 a set @ ¦ 9 union ¡ and ; often < , is ¨ = a is > collection > ; such . That ¦ that - & is, ¦ & of & a disjoint ¦ D partition .) ; Theorem 2: Let be an equivalence relation on a set . Then the equivalence classes of form a partition of ; . ¡ > = ; ¨ < Conversely, given a partition of the set , there > ¡ = ¨ is an equivalence relation that has the sets < , , as its equivalence classes. Proof (informal): The equivalence classes of an equivalence ¤ ¤ ¨ % relation are nonempty (since # $ ), and by Theorem 1 are disjoint. Since every element of the set ; is in some ¤ ¤ ¨ % equivalence class (e.g. # $ ), the equivalence classes partition ; . 10 the Define No some Example: (Proof w equi , assume = . of a v It relation alence Theorem is W not e we EGF can E E E7X E U Z P classes +JI H hard +\I +JI +JI +SI H H H H on ha partition 2, v ; e K7L to K7L K7L K7L K7L continued) L a by L L L L L modulo L L L L see partition ¤ MON M M M M N N N N V [ that the P Y T F MON MON MON MON 11 M N if 2 set X this U Z P and Q M M M M as X M U Z P F ¡ of M M MW[ MWV is M follo < Y T only Q inte M M M M M an P P P P ¨ = X P U Z F P ws: L L L L M M M M gers equi L M if L L L L L L L L L > ¤ L R R7L R R R ¦ v M M M according of M ¨ alence a ¡ set < for relation ; . to w English Example: same ords letter as w follo ords _ ¤ # #ed # as Let = $ ^ f ¤` ws: $ defined . $ Then be & & & the we by ¤ d equi ^ ^ ¦ _ can ¤ ¦ D] ¤ v = 12 a alence partition ¦ ¦ ¤ d ¦ _ if ` D ^ and ^ ` ¤ ¦ d ¦ ¤ relation = only ¦ _ f ^ the ¦ d & ¦ bc b & if set & b D & ¦ on & & ¦ of & ¦ & starts & the & English ¦ set with of the.