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Are the following 3 statements true and does the cantorbernstein theorem followby Wiz14
Tags: cantor set theory 
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#1
Nov2312, 03:46 AM

P: 20

1.There exists an injection from A to B ⇔ A ≤ B
2.There exists an injection from B to A ⇔ B ≤ A 3.If A ≤ B and B ≤ A, then A = B Does this prove the Cantor Bernstein theorem? Which says that if 1 and 2 then there exists a Bijection between A and B (A = B) And if it does, why is there a different, longer proof for it? 


#2
Nov2312, 04:35 AM

P: 144




#3
Nov2312, 01:43 PM

P: 1,583

Statement 3 IS the CantorSchroederBerstein theorem: "If the cardinality of A is less than or equal to the cardinality of B, and the cardinality of B is less than or equal to the cardinality of A, then the cardinality of A is equal to the cardinality of B." You can also state it as "If there is an injection from A to B, and there is an injection from B to A, then there is a bijection from A to B." As Norweigan said, it requires a nontrivial argument to prove this theorem.
EDIT: See the easytounderstand proof here. 


#4
Nov2312, 02:03 PM

P: 20

Are the following 3 statements true and does the cantorbernstein theorem follow
A ≤B and B ≤ A is like saying A = B or A is strictly less than B and B is strictly less than A, which is a contradiction, so A must = B. 


#5
Nov2312, 05:20 PM

P: 1,583




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