This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Cardinality Part II Original Notes adopted from January 8, 2002 (Week 14) c P. Rosenthal , MAT246Y1, University of Toronto, Department of Mathematics Definition.  S  =  T  if there exists f : S → T (onetoone and onto) Eg .  Even naturals  =  natural  Definition. If  S  =  N  , then S is countably infinite (or denumerable , or enumerable ) and  S  = ℵ (aleph nought). [Elements of S can be listed]  Q  = ℵ Theorem. Union of countable number of countable sets is countable. [0,1] is uncountable  [0 , 1]  = c ,  R  = c Definition.  S  6  T  if there exists T ⊂ T such that  S 1  =  T  . ie) There exists f : S → T such that f is onetoone (not necessarily onto). SchroederBernstein(Cantor) Theorem If  S  6  T  and  T  6  S  , then  S  =  T  . Proof : Given f : S 1:1 → T , g : T 1:1 → S , we want h : S → T (onetoone and onto). For s ∈ S , a first ancestor is a t ∈ T such that g ( t ) = s . (If s / ∈ g ( T ), no first ancestor.) A...
View
Full
Document
This note was uploaded on 04/26/2011 for the course MATH 246 taught by Professor Applebaugh during the Spring '10 term at University of Toronto.
 Spring '10
 Applebaugh
 Math

Click to edit the document details