Proving Power Set Inclusion When B is Included in C

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To prove that if set B is included in set C, then the power set of B is included in the power set of C, one can start with the definitions of power sets and subsets. The power set of C contains all subsets of C, which inherently includes all subsets of any subset of C, including B. Therefore, every subset of B, which constitutes the power set of B, is also a subset of C. This leads to the conclusion that the power set of B must be included in the power set of C. The argument is formal and aligns with set theory principles.
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How do you show that if B is included in C, then the power set of B is included in the power set of C?
 
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What have you tried already? I'd imagine just starting with the definitions would give a strong clue.
 
Power set of C includes all subsets of C, which means that it also includes all subsets of a subset (B). Thus it includes the power set of B.
Is this formal enough?
 

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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