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Obviously
Post  Post subject: Defining infinite sets  |  Posted: Mon Jan 20, 2014 4:00 pm
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I'm having some troubles doing this. The exercise is as follows:

Given two infinite sets:
S = { 5,10,15,20,25,... } and
T = { 3,4,7,8,11,12,15,16,19,20,... }.
Specify each of these sets by defining the properties Ps and Pt such that S = { x ∈ N : Ps(x) } and T = { x ∈ N : Pt(x) }.

N is natural numbers btw.

I tried defining the property of S like this:
Ps = { x ∈ S : x ∪ |{0,1,2,3,4}| }

This looks retarded, to put it bluntly, but I'm not sure what else I can do. I suppose I should instead try and define something like "every fifth element in N". Not sure how to go about this though. The book doesn't help.

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marnixR
Post  Post subject: Re: Defining infinite sets  |  Posted: Mon Jan 20, 2014 5:29 pm
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sorry i can't help - this is so totally over my head

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Obviously
Post  Post subject: Re: Defining infinite sets  |  Posted: Tue Jan 21, 2014 1:55 pm
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I suddenly got the idea that I could use the difference in sets and came up with this:
Ps = { x ∈ S : x ∉ {1,2,3,4,6,7,8,9} }

I quickly realized this wouldn't do either.

I really hope I survive this semester... :roll:

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Prometheus
Post  Post subject: Re: Defining infinite sets  |  Posted: Tue Jan 21, 2014 5:04 pm
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Could you define S as something like the multiples of 5 of all the positive natural numbers? So N=1 would map to x=5, N=2 would map to x=10 and so on. Not sure how that would look notationally though, something like:

S = { x ∈ N : 5N for all positive N }


I know how you feel by-the-way. Who would have thought this maths stuff was so hard, hey?


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Obviously
Post  Post subject: Re: Defining infinite sets  |  Posted: Wed Jan 22, 2014 9:14 am
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Prometheus wrote:
I know how you feel by-the-way. Who would have thought this maths stuff was so hard, hey?


I know this is supposed to be easy, but for some reason my mind gets stuck on trivial things. I thought of doing this through a bijection or something like you seem to suggest yet for whatever reason I thought that was not how I was supposed to solve it.

Prometheus wrote:
S = { x ∈ N : 5N for all positive N }


I suppose something more is needed. Like having a function f : S --> N. Then something like f(s) --> n or whatever. Then having defining the function as being a property of S. I'll have to play around with it a little later I suppose.

Thanks for the input!

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Obviously
Post  Post subject: Re: Defining infinite sets  |  Posted: Thu Jan 23, 2014 7:51 pm
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Obviously wrote:
Prometheus wrote:
I know how you feel by-the-way. Who would have thought this maths stuff was so hard, hey?


I know this is supposed to be easy, but for some reason my mind gets stuck on trivial things. I thought of doing this through a bijection or something like you seem to suggest yet for whatever reason I thought that was not how I was supposed to solve it.

Prometheus wrote:
S = { x ∈ N : 5N for all positive N }


I suppose something more is needed. Like having a function f : S --> N. Then something like f(s) --> n or whatever. Then having defining the function as being a property of S. I'll have to play around with it a little later I suppose.

Thanks for the input!


EDIT:
Turns out I was overcomplicating things (as is typical for me). It should suffice with something like Ps = { x ∈ S : 5x ∈ N } I think. Oh well.

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iNow
Post  Post subject: Re: Defining infinite sets  |  Posted: Thu Jan 23, 2014 11:51 pm
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Obviously wrote:
Turns out I was overcomplicating things...

Story of my life. :lol:

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shlunka
Post  Post subject: Re: Defining infinite sets  |  Posted: Wed Sep 28, 2016 4:20 pm

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I tried Kolmogorov complexity but it didn't work.

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Olinguito
Post  Post subject: Re: Defining infinite sets  |  Posted: Sun Jan 01, 2017 6:42 pm
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My preferred definition of an infinite set:

    A set X is infinite iff it has a proper subset Y such that a bijection f:XY exists.
However, the equivalence of this to other definitions of infinite sets may depend on the axiom of choice in Zermelo–Fraenkel set theory, on which modern mathematics is based.

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