Proof by Induction: Closure of Union = Union of Closures

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SUMMARY

The discussion centers on the validity of using proof by induction to demonstrate that the closure of a union of finite sets equals the union of the closures of those sets. Participants express skepticism about the use of induction without a presented proof, emphasizing the importance of logical rigor in any proof method. While some prefer direct proofs over induction, the consensus is that as long as the logic is sound, the proof is valid. The conversation highlights a divergence in mathematical proof preferences, particularly regarding finite sets.

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  • Understanding of mathematical induction
  • Familiarity with set theory and closures
  • Knowledge of finite sets
  • Basic principles of logical reasoning in proofs
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  • Research the principles of mathematical induction in depth
  • Study set theory, focusing on closures and their properties
  • Explore alternative proof methods beyond induction
  • Examine the Brouwer–Hilbert controversy for insights into proof preferences
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Mathematicians, students of mathematics, and anyone interested in the foundations of proof techniques in set theory and mathematical logic.

Bachelier
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Is it a solid proof to show that the closure of a union equals the union of the closures of the sets via induction?
 
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Bachelier said:
Is it a solid proof to show that the closure of a union equals the union of the closures of the sets via induction?

How can we possibly know if your proof is valid if you don't show us your proof? Still, I'm not entirely sure why you would use induction here. Are you only considering the union of finitely many sets?
 
A proof is a proof no matter if you use induction or some direct method as long as the logic of each step is impeccable.

There are some math purists who disdain induction proofs for some theorems and prefer other styles of proof but that's beyond my math understanding to explain here:

http://en.wikipedia.org/wiki/Brouwer–Hilbert_controversy
 
Number Nine said:
How can we possibly know if your proof is valid if you don't show us your proof? Still, I'm not entirely sure why you would use induction here. Are you only considering the union of finitely many sets?

Yes of course. I'm talking about finite sets. I found a different way to prove it though. But the proof by induction crossed my mind.

I should work it out some time and post it here. It seems feasible.
 
jedishrfu said:
A proof is a proof no matter if you use induction or some direct method as long as the logic of each step is impeccable.

There are some math purists who disdain induction proofs for some theorems and prefer other styles of proof but that's beyond my math understanding to explain here:

http://en.wikipedia.org/wiki/Brouwer–Hilbert_controversy

great article. Thanks
 

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