Set Theory question.

[Cogman]

Ok, so maybe this is more sequential theory.

So before the question, consider this

If I have an infinite set, A, that is defined by the function
f(n) = 2n

And another set, B, defined by the function
f(n) = n

I can look at that (and hopefully you can too) and easily say that set A is a subset of set B. You can also say that the intersection of Set A with Set B will equal Set B, and the union of A and B equals A.. Ect.

So the question is, is there a way to determine unions, intersections, and subsets for more complex function based sets? Like, for example,

Set A: f(n) = 2n^2 + 2n
Set B: f(n) = 3n^2 + 2

 

[CP5670]

Yes, you can do those operations in the obvious way. The one thing that becomes tricky is determining whether two infinite sets are the same or equivalent in some way, since there is no one notion of "sameness" that works in all situations.

 

[Cogman]

Originally posted by: CP5670
Yes, you can do those operations in the obvious way. The one thing that becomes tricky is determining whether two infinite sets are the same or equivalent in some way, since there is no one notion of "sameness" that works in all situations.

Understood, but I guess I should clarify. Is there anyway to determine what function will be dominate, or what parts of a function to use when preforming union, intersections, ect of a set?

 

[CTho9305]

If I have an infinite set, A, that is defined by the function
f(n) = 2n

And another set, B, defined by the function
f(n) = n

I can look at that (and hopefully you can too) and easily say that set A is a subset of set B.

Are you claiming that there are more integers than there are even integers? There aren't...

 

[Cogman]

Originally posted by: CTho9305

If I have an infinite set, A, that is defined by the function
f(n) = 2n

And another set, B, defined by the function
f(n) = n

I can look at that (and hopefully you can too) and easily say that set A is a subset of set B.

Are you claiming that there are more integers than there are even integers? There aren't...

Yes, I realize that , that wasn't what is implied though. There are no even numbers that aren't integers, therefore it is accurate to say the a set of even integers is a subset of integer. It is not correct to say that a set of even integers is bigger then a set of integers.

 

[CP5670]

Understood, but I guess I should clarify. Is there anyway to determine what function will be dominate, or what parts of a function to use when preforming union, intersections, ect of a set?

If you actually want to calculate those things for specific sets, it means finding integer solutions to systems of equations, which in general is very hard. You can only do that easily in some simple cases like linear functions.

Are you claiming that there are more integers than there are even integers? There aren't...

There may be, depending on what "more" means in the context. There is a one-to-one correspondence between them, but at the same time the even integers have a lower asymptotic density.