Proving Power Set Inclusion When B is Included in C

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SUMMARY

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 must 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 B. Therefore, the power set of B is indeed a subset of the power set of C, establishing the inclusion definitively.

PREREQUISITES
  • Understanding of set theory concepts, particularly subsets and power sets.
  • Familiarity with mathematical notation and definitions related to inclusion.
  • Basic knowledge of logical reasoning and proofs.
  • Experience with formal mathematical writing and argumentation.
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  • Study the definitions and properties of power sets in set theory.
  • Explore examples of subset relationships and their implications on power sets.
  • Learn about formal proof techniques, such as direct proof and proof by contradiction.
  • Investigate related concepts, such as Cartesian products and their power sets.
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Mathematicians, students studying set theory, and anyone interested in formal proofs and logical reasoning in mathematics.

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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?
 

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