Write this in fopa lang?

• Apr 30th 2011, 06:26 AM
ikurwae89
Write this in fopa lang?
if k is any natural numb, then n^3>kn^2 for all sufficent large n.
• Apr 30th 2011, 06:32 AM
Plato
Quote:

Originally Posted by ikurwae89
if k is any natural numb, then n^3>kn^2 for all sufficent large n.

What does "Write this in fopa lang?" mean?
• Apr 30th 2011, 07:49 PM
ikurwae89
first order penos arithmetic
• May 2nd 2011, 04:41 AM
emakarov
$\forall k\exists m\forall n.\,n\ge m\to n^3>kn^2$

If >= is not considered a part of the language of PA, then it can be replaced using the following equivalence.

$\forall n\forall m.\,n\ge m\leftrightarrow\exists p.\,n=m+p$
• May 3rd 2011, 08:24 PM
ikurwae89
can u explain to me how u got that?

This is what i got.

((For all k)( there exists m)( there exists n)(n.n.n > k.m.m.m))
i dont have the upside A and E.

sorry

what is the difference?
• May 4th 2011, 01:50 AM
emakarov
Quote:

((For all k)( there exists m)( there exists n)(n.n.n > k.m.m.m))
(1) The original problem says "for all sufficent large n," while your formula says "there exists n."

(2) I don't understand why you replaced "n^3 > kn^2" with "n^3 > km^3."

(3) "For all sufficiently large n" means "for all n that exceed a certain lower bound m." That's why I said $\exists m\forall n.\,n>m\to\dots$ (one can also use >= instead of > here).
• May 5th 2011, 03:59 AM
ikurwae89
sorry about my stuff up and thanks for making the for all sufficient large n clear :)

so this is how you would write it.

((For all k)( there exists m)( for all n)(n>=m)--> (n.n.n > k.n.n.))

Thanks a lot btw really helpful :)
• May 5th 2011, 05:14 AM
emakarov
Quote:

so this is how you would write it.

((For all k)( there exists m)( for all n)(n>=m)--> (n.n.n > k.n.n.))
Yes.