Just shortly (but, please tell me how you know):
When I say "all natural numbers up to n" does that include (or exclude) n?

Question

astrophysics · Answer

include

idku · Answer

How do you know? (I'm writing proof, just trying to be precise...)
Is that supposed to be one of those common sense facts?

astrophysics · Answer

Yeah I think it's a common notion

astrophysics · Answer

Not sure though

jhonyy9 · Answer

n is just the name of this line of natural numbers 
just bc. than you write (n+1) so this mean natural numbers too 
or than you write n+2 this mean natural numbers too

idku · Answer

Oh, I don't mean to say $\mathbb{N}$ by my lower case n, if that is what you are referring to.

idku · Answer

I mean this:

When you say "all natural number up to x" which (of the below) is it.
a.    ${\rm all~natural~numbers~up~to}~x=~\{1,~2,~3,~~...~~(x-2),~(x-1),~x\}$
b.    ${\rm all~natural~numbers~up~to}~x=~\{1,~2,~3,~~...~~(x-2),~x-1\}$

idku · Answer

Like if I want to include all natural numbers from 1 to x, would it be correct to say "all natural numbers up to x"(?)

jhonyy9 · Answer

yes this include x sure

idku · Answer

Alright, thanks for the confirmation ... it does sounds like it includes x, but, just want to check my language.

jhonyy9 · Answer

<< ... up to x >> this mean that the greater is x

idku · Answer

"greatest" you mean?

jhonyy9 · Answer

yes exactly

idku · Answer

Alright:) tnx:)

jhonyy9 · Answer

np

was my pleasure 

anytime 

good luck

idku · Answer

I select $n_p$.

jhonyy9 · Answer

np mean no problem

idku · Answer

I know, it's a joke about primes:)