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Interesting induction questions

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RuiAce:
Warning: What follows gone tremendously outside "Mathematics Extension 2" territory. Concepts here are NOT examinable and should NOT be treated as the standard of the course. Technically, not even set theory is examinable.

Disclaimer: Not all my own material. Credit shared to someone who helped me out. Which leads to query - what is the source of this question? Was it actually doable by 4U methods

Note that existential quantification is assumed at the start as per the wording of the question. If universal quantification were chosen, then for n=1, take the set {1, 2, 3, 4}. Then clearly 1+2 = 3 and 3 is not divisible by 2.



By extension, it can be checked that P(0) is true as every integer is divisible by 1.
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Ali_Abbas:
I believe I may have a simpler, non-inductive proof of the result.

RuiAce:
I only thought in that direction because it said induction. Although nice seeing that exhaustion is 'somewhat' tidy, because usually it's just messy

Small technicalities:
- It is bounded below by 2n but I assumed that the elements must be distinct, so it's actually bounded below by 1+2+...+2n. This doesn't matter too much.
- That condition on the even number of odds and evens requires clarification. That is still a lemma; it's intuitive but not obvious.

Mahan:

--- Quote from: RuiAce on November 01, 2016, 06:28:00 am ---Warning: What follows gone tremendously outside "Mathematics Extension 2" territory. Concepts here are NOT examinable and should NOT be treated as the standard of the course. Technically, not even set theory is examinable.

Disclaimer: Not all my own material. Credit shared to someone who helped me out. Which leads to query - what is the source of this question? Was it actually doable by 4U methods

Note that existential quantification is assumed at the start as per the wording of the question. If universal quantification were chosen, then for n=1, take the set {1, 2, 3, 4}. Then clearly 1+2 = 3 and 3 is not divisible by 2.



By extension, it can be checked that P(0) is true as every integer is divisible by 1.
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--- End quote ---

Thank you RuiAce, that was an interesting proof but I think we can come up with a more simple and a proof of at extension2 level.The interesting thing about this proof is we use induction hypothesis multiple times whereas in usual induction proof we use induction hypothesis only once.Also on the prod you don't need the distinction assumption.
Here is the proof:
 

RuiAce:
Whilst I don't deny the course has been made easier, neither of the proofs provided are to within reasonable bounds of the current Extension 2 level, especially with how the inductive hypothesis must be applied more than once.

Because expecting an Extension 2 student to think that deeply into pure mathematics is stretching too far. The necessity of considering the elements in such a way, despite being one of the easiest things to teach, is not something that one is expected to know.

As for simplicity though, I'd say that your answer is indeed simpler

(We could really add a beyond 4U section of this forum but I didn't bring it up cause I didn't see enough value in it yet - forum still needs popularity for that.)
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Also, I still wish to ask for my friend. What is the source of this question?

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