Equivalence between a set and the subset of its subset

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The discussion centers on the equivalence relation between a set A and its subset A1, specifically questioning whether A's lack of equivalence to A1 implies A's lack of equivalence to any subset of A1. It is established that the truth of this statement depends on the chosen equivalence relation, such as bijection or other criteria like parity. When using bijections, the statement holds true; however, under different equivalence relations, it may not. The conversation reflects on the complexities of understanding these relationships, referencing Cantor-Bernstein's theorem as a learning experience. Ultimately, the nuances of equivalence relations significantly influence the validity of the original claim.
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Is it true that: If A is not equivalent to its subset A1. Then A is not equivalent to any subset of A1?
 
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You should consider also informing us about the relevant equivalence relation, as the truth or falsity of your statement heavily depends on that information.

For example, if you for your eq.rel. use existence of a bijection, the statement is true, but other relations, like having same parity, will render your statement false.
 
actually this is trivial. I just learned all this the complicated way, .i.e. proving cantor bernstein without the well ordering theorem.
 

Thread 'Deductive proof in logic formal systems'

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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