Are Finite Sets and the Set of All Integers in R^2 Closed?

[jwqwerty]

i have one simple question

if we a consider subsets of R^2 which are: a finite set and set of all integers,
then aren't a finite set and set of all integers not closed? For instance for set of all integers, it do not have any limit points. thus by definition of closed (E is closed if all limit points of E are points of E) the set is not closed. Or did i do something wrong?

 

[Alesak]

Yes, but the sequence which would converge to certain limit in E point need to be composed from points that are in E. So the sequences you need to consider are composed entirely of integers.

 

[Citan Uzuki]

If a set has no limit points, then it is vacuously true that all of the limit points are in the set, so the set is closed.

 

[jwqwerty]

If a set has no limit points, then it is vacuously true that all of the limit points are in the set, so the set is closed.

Sorry but could you tell me further about 'vacuously true'? I have never encountered that word. Thank you.

 

[Citan Uzuki]

It means the statement is true because it doesn't actually say anything. If P is any property whatsoever, it is vacuously true that \forall x \in \emptyset, P(x). The reason why this statement is true is usually illustrated by looking at its negation -- if it were not true, then there would be some x in the empty set for which P(x) does not hold. But that is impossible, since there are no x in the empty set.

As applied to this particular problem, if E is a set with no limit points, then E will be closed, because for E not to be closed, there would have to be a limit point of E which is not in E. But there are no limit points of E, in E or otherwise. So E is closed.

 

[zmodnz]

Citan...nice try.

I cannot tell much about limit points and stuff...but i can tell about elementary mathematical logic...Ha.

I just took about 15 minutes typing slowly to show you why you cannot use 'that reasoning' here...

but when i hit 'submit reply'...guess what...Yeap...i lost all my hard work...i wasn't connected...typing is slow and quite frustrating for me.

I'll retype the situation again tomorrow.

 

[SteveL27]

i have one simple question

if we a consider subsets of R^2 which are: a finite set and set of all integers,
then aren't a finite set and set of all integers not closed? For instance for set of all integers, it do not have any limit points. thus by definition of closed (E is closed if all limit points of E are points of E) the set is not closed. Or did i do something wrong?

The complement of the integers in R^2 is an open set. So is the complement of a finite set.

 

[zmodnz]

Again i just lost about 20min worth of hard typing.

i was connected when i started typing and by the time i had finished guess what?

i wasn't connected any more.

 

[zmodnz]

So i'll do an injustice to my post by just saying.

Citan is correct by a Mathematical convention about the use of the null set.

But, in Logic this convention's use is quite stricter...i.e...many things have to be satisfied before it can be used...unlike as in Mathematics.

Example...P(x) has to be a logically valid formula.

...etc...
In Logic the convention says that:

every interpretation of a formal language is a model of the empty set.Sorry for the Philosophy Detour guys.
Frege...Russell and The Boys...Ha.

 

[SteveL27]

Again i just lost about 20min worth of hard typing.

i was connected when i started typing and by the time i had finished guess what?

i wasn't connected any more.

When you type a long post into a web form (on this or any other site), do a Copy before hitting Submit. That way when something goes wrong, you still have your post. After a while this will become second nature.

 

[wisvuze]

So i'll do an injustice to my post by just saying.

Citan is correct by a Mathematical convention about the use of the null set.

But, in Logic this convention's use is quite stricter...i.e...many things have to be satisfied before it can be used...unlike as in Mathematics.

Example...P(x) has to be a logically valid formula.

...etc...



In Logic the convention says that:

every interpretation of a formal language is a model of the empty set.


Sorry for the Philosophy Detour guys.
Frege...Russell and The Boys...Ha.

Huh?? Vacuous truths are also "allowed" in "regular logic".. What is so philosophical about this?

 

[zmodnz]

No, i wasn't using philosophical to describe 'vacuous truths.

I was using philosophical to describe what many many (almost all) modern mathematicians call Logic....i wonder if the Intuitionist 'believes in' the convention of vacuous truth...i wouldn't be surprised if they don't...ha.

 

[zmodnz]

Oh yeah!

Thanks for reminding me that we can copy & paste, Steve.

I forgot about that.

Fancy that.