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Purpose of proof by minimum counter example?

So, my notes essentially have a proof;

defined are polynomials f0(x)=1,f1(x)=x,fn(x)=2xfn−1(x)−fn−2(x)(n=2,3,...) f_0(x) =1, f_1(x) = x, f_n(x) = 2xf_{n-1}(x) - f_{n-2}(x) (n=2,3,...)

let x∈R x \in \mathbb{R}

Let P(n) be the statement fn(−x)=(−1)nfn(x) f_n(-x) =(-1)^nf_n(x)

"f0 f_0 and f1f_1 proven"

Assume p(n) is not true

Let minimal criminal be 2

Then it shows that; fn(−x)=(−1)nfn(x) f_n(-x) = (-1)^nf_n(x) using the definitions given andp(n−1),p(n−2) p(n-1), p(n-2) .

This contradicts the assumption that p(n) is false so p(n) is true.

(some intermediate steps skipped above to make the post shorter).

But what is the point of assuming that p(n) is false just to prove that p(n) is true if it can be proven true without an assumption of the negation with "strong induction " anyway? It seems that the method of induction would make proof by minimum counter example redundant.
(edited 6 years ago)
The two concepts are very closely related. To be honest, I prefer "smallest counterexample" to induction in many ways. We generally spend several hours on explaining how/why induction works, but we usually don't even bother explaining smallest counterexample at all, because it's kind of obvious that it works.

You could argue that it's induction that's the redundant concept...
Original post by DFranklin
The two concepts are very closely related. To be honest, I prefer "smallest counterexample" to induction in many ways. We generally spend several hours on explaining how/why induction works, but we usually don't even bother explaining smallest counterexample at all, because it's kind of obvious that it works.

You could argue that it's induction that's the redundant concept...


Thanks for the reply. I guess you could argue that, but I just thought about it the other way around because I learnt standard proof by induction first.

Could I ask why you prefer smallest counterexample?
Original post by NotNotBatman
Thanks for the reply. I guess you could argue that, but I just thought about it the other way around because I learnt standard proof by induction first.

Could I ask why you prefer smallest counterexample?


Say you have some rules, you know are true
By assuming these rules are false, you can show a lot of assertions but they end up implying something that contradicts your original rules.
Therefore the assertions are false, and if everything the rules being false implies is false, then the rules must be true.
(edited 6 years ago)
Original post by emperorCode
Say you have some rules, you know are true
By assuming these rules are false, you can show a lot of assertions but they end up implying something that contradicts your original rules.
Therefore the assertions are false.


Ok, so are you saying that it's easier to disproof the negation of "if p(n) then p(n+1)" like in some other standard implications?

This is the only example I've ever looked at of the minimum counterexample, so I haven't come across those types yet.
Original post by NotNotBatman
Thanks for the reply. I guess you could argue that, but I just thought about it the other way around because I learnt standard proof by induction first.

Could I ask why you prefer smallest counterexample?
I thought I explained it in my previous post. People generally find it easier to understand, you don't have all the formal "boiler plate" to learn (All that "assume true for n=k" stuff) like you do for induction. And you don't have to invoke a "magic" principle ("by the principle of mathematical induction" - it makes me think of He-man saying "by the power of Grey skull"!) to justify the final conclusion.
Original post by NotNotBatman
Ok, so are you saying that it's easier to disproof the negation of "if p(n) then p(n+1)" like in some other standard implications?

This is the only example I've ever looked at of the minimum counterexample, so I haven't come across those types yet.


Exactly.
You're trying to prove those rules being false makes no sense
Original post by DFranklin
I thought I explained it in my previous post. People generally find it easier to understand, you don't have all the formal "boiler plate" to learn (All that "assume true for n=k" stuff) like you do for induction. And you don't have to invoke a "magic" principle ("by the principle of mathematical induction" - it makes me think of He-man saying "by the power of Grey skull"!) to justify the final conclusion.


Thank you - had a chuckle because of the he man reference too.

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