# Thread: Write this in fopa lang?

1. ## Write this in fopa lang?

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

2. 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?

3. first order penos arithmetic

4. $\displaystyle \forall k\exists m\forall n.\,n\ge m\to n^3>kn^2$

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

$\displaystyle \forall n\forall m.\,n\ge m\leftrightarrow\exists p.\,n=m+p$

5. 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?

6. ((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 $\displaystyle \exists m\forall n.\,n>m\to\dots$ (one can also use >= instead of > here).

7. 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

8. 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.