Are the following 3 statements true and does the cantor-bernstein theorem follow

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The discussion centers on the validity of three statements regarding set cardinalities and their implications for the Cantor-Bernstein theorem. It clarifies that while statements one and two define cardinality relationships through injections, statement three, which asserts that if A ≤ B and B ≤ A then A = B, is essentially the Cantor-Schroeder-Bernstein theorem. The theorem requires a nontrivial proof, as the properties of cardinalities in sets do not directly mirror those of numbers. A common misconception is addressed, emphasizing that the relationships among sets cannot be assumed to behave like numerical inequalities. Ultimately, the proof of the Cantor-Bernstein theorem is necessary to establish the equivalence of cardinalities in a rigorous manner.
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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?
 
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Wiz14 said:
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?

No. 1) and 2) are definitions of ≤ and ≥ for cardinalities of sets. 3) is a nontrivial consequence for which you have provided no argument at all.
 
Statement 3 IS the Cantor-Schroeder-Berstein 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 easy-to-understand proof here.
 
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lugita15 said:
Statement 3 IS the Cantor-Schroeder-Berstein 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 easy-to-understand proof here.

I am reading that proof now but where is the flaw in my reasoning?
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.
 
Wiz14 said:
I am reading that proof now but where is the flaw in my reasoning?
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.
If A and B were numbers, then yes it would be trivially true that if A≤B and B≤A then A would equal B. But A and B are sets, and what we mean by A≤B is that "there exists an injection from A to B". We don't know beforehand whether "less than or equal to" for sets, which has to do with existence of an injection, has the same properties as "less than or equal to" for numbers. We have to prove it. So you can't use your familiar properties of numbers, like the fact that two things can't be strictly less than each other.
 
I was reading documentation about the soundness and completeness of logic formal systems. Consider the following $$\vdash_S \phi$$ where ##S## is the proof-system making part the formal system and ##\phi## is a wff (well formed formula) of the formal language. Note the blank on left of the turnstile symbol ##\vdash_S##, as far as I can tell it actually represents the empty set. So what does it mean ? I guess it actually means ##\phi## is a theorem of the formal system, i.e. there is a...
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