Proof by Induction: Closure of Union = Union of Closures

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Discussion Overview

The discussion revolves around the validity of using proof by induction to demonstrate that the closure of a union of sets is equal to the union of the closures of those sets. The scope includes theoretical aspects of mathematical proofs and the nature of different proof techniques.

Discussion Character

  • Debate/contested

Main Points Raised

  • Some participants question the validity of the proof without seeing it, particularly asking if the induction is being applied to finitely many sets.
  • Others assert that any proof, whether by induction or direct methods, is valid as long as the logic is sound, though they acknowledge differing opinions on the use of induction in mathematics.
  • A participant mentions having found an alternative proof method but still considers induction as a feasible approach.
  • References to the Brouwer–Hilbert controversy are made to illustrate differing perspectives on proof methods in mathematics.

Areas of Agreement / Disagreement

Participants express differing views on the appropriateness of using induction for this proof. There is no consensus on whether induction is the best method, and some participants emphasize the importance of logical rigor over the method used.

Contextual Notes

Some assumptions about the nature of the sets involved (finite vs. infinite) are not fully clarified, and the discussion does not resolve whether induction is suitable for all cases.

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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