set-theory+ - 1 2 as an index variable let γ f 1,f 2:= h n...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
Set theory Miscellany Jonathan L.F. King University of Florida, Gainesville FL 32611-2082, USA [email protected] Webpage http://www.math.ufl.edu/ squash/ 1 May, 2006 (at 02:10 ) Abstract: A NaPo conundrum. Meir’s Finite inter- section problem. Pre-image problem. Unfinished: Ordinal arithmetic. Cardinal arithmetic. Not yet: Hercules and the Hydra. Problem V4 from a NaPo exam Below, A,P,Q are arbitrary sets. Soln- V4a: Here is a bijection H : A P × Q , ± [ A P ] Q . Let H ( f ) := b f , where b f ( q ) ( p ) := f ( p,q ). Or in one swell foop, H ( f ) := ± q 7→ [ p 7→ f ( p,q )] ² . b Below, I need some two 2-element sets; let’s take { 1 , 2 } and {♥ , ♣} . Use 2 B to abbrevi- ate {♥ , ♣} B ; the set of maps from B →{♥ , ♣} . I am given bijections P : N , ± N ×{ 1 , 2 } and Q : R , ± 2 N , and the inverse-fnc G := Q 1 mapping 2 N , ± R . ITOf P , Q , G , I want to define these bijections: β : R 2 , ± 2 N × 2 N . 1 : γ : 2 N × 2 N , ± 2 N ×{ 1 , 2 } . 2 : δ : 2 N ×{ 1 , 2 } , ± 2 N . 3 : I’ll then combine them to produce a bijection ε : R 2 , ± R . 4 : Soln- V4b: Well, R 2 is R × R , so it works to define β ( x 1 ,x 2 ) := ( Q ( x 1 ) , Q ( x 2 ) ) . 1 : Using f for a general element of 2 N , we can write ( f 1 ,f 2 ) for a gen-elt of 2 N × 2 N . Employing j ∈ {
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 1 , 2 } as an index variable, let γ ( f 1 ,f 2 ) := h ( n, j) 7→ f j ( n ) i . ‡ 2 : For m a natnum, the value P ( m ) is a natnum-index pair. Write this pair as ( m N ,m Idx ) . For example: If P (5177) = ( 38 , 2) , then 5177 N = 38 and 5177 Idx = 2 . Use h for a general fnc in 2 N ×{ 1 , 2 } . Let δ ( h ) := ± m 7→ h ( m N ,m Idx ) ² note ==== h ◦ P . ‡ 3 : This δ is indeed a bijection 2 N ×{ 1 , 2 } , ± 2 N . Composing functions. Here is γ ◦ β written out in full: γ ( β ( x 1 ,x 2 ) ) = h ( n, j) 7→ Q ( x j ) ( n ) i . So δ ◦ γ ◦ β maps ( x 1 ,x 2 ) to ± m 7→ Q ( x m Idx ) ( m N ) ² . Defining ε := G ◦ δ ◦ γ ◦ β produces ε ( x 1 ,x 2 ) = G ³ m 7→ Q ( x m Idx ) ( m N ) ´ . Filename: Problems/SetTheory/set-theory+.latex As of: Thursday 27Apr2006 . Typeset: 1May2006 at 02:10 . 1...
View Full Document

This note was uploaded on 01/26/2012 for the course MAS 3300 taught by Professor Staff during the Summer '08 term at University of Florida.

Ask a homework question - tutors are online