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#1 2013-10-16 04:10:22

Reuel
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Set notation for corresponding sets

Hi!

I don't know a lot about sets or set notation but I have taught myself the basics. I was wondering if anyone could please tell me how to formally make the statement that the members of two sets directly correspond with each other.

For instance, if we have two arbitrary sets




and we wish to make the statement that 0 corresponds with a, 1 corresponds with b, etc., how would we go about doing that?

Also, they are not the same set. A real world example might be moments in time and events in time where each event occurs in each moment. What would be the formal mathematical expression?

Thanks! smile

#2 2013-10-16 04:23:38

bob bundy
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Re: Set notation for corresponding sets

I'd do it like this:



The backwards E stands for "there exists".

A bijection is a relationship that is 1 to 1 between the sets.

http://www.mathsisfun.com/sets/injectiv … ctive.html

http://en.wikipedia.org/wiki/Bijection

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

#3 2013-10-16 04:56:55

Reuel
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Re: Set notation for corresponding sets

Thank you! That is exactly what I was looking for.

Last edited by Reuel (2013-10-18 23:48:48)

#4 2013-10-18 23:48:58

Reuel
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Re: Set notation for corresponding sets

In the same fashion, if we have a number line made up of all the real numbers and if there is a boundary within



how do we say that we want to have the set of everything from a to b in terms of all reals minus everything except for what is in the set from a to b? I understand the set from a to b is a subset of all reals but how can it be defined as the set of all reals minus everything except what exists within the set a->b?

On a related note, does anyone know of a good web site or book that teaches all about set notations? I checked out Khan Academy but there were only a few videos on the basics.

Thanks.

#5 2013-10-19 00:42:14

anonimnystefy
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Re: Set notation for corresponding sets

You, mean

Last edited by anonimnystefy (2013-10-19 00:42:44)


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#6 2013-10-19 01:12:16

Reuel
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Re: Set notation for corresponding sets

Yes, thank you.

Are there any good books on set notation? I want to learn to write more formally in mathematics but the web sites I have looked at thus far are not so great.

Thanks again.

Actually you probably meant



but I knew what you meant.

Last edited by Reuel (2013-10-19 01:17:14)

#7 2013-10-23 04:20:07

Reuel
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Re: Set notation for corresponding sets

Okay, here is what I have. I would appreciate any edits or suggestions anyone who knows sets or mathematical writings might have to offer to help improve the quality of what I have so far.

---

If time is initially defined as a well-ordered uncountable set



[in that it might be said to go from ±∞], the set T might be written so as to contain some boundary condition



so that it can be rewritten as

;

but if we further define the set as being of certain relevance to only that boundary condition Z, we say let



so as to be redefined as

.

Suppose then we have a series of events in time that are also restricted to Z (which is actually what I intend to be the reason for why Z is a boundary condition in the first place) and let that ordered set (ordered because it corresponds to time T) be given by

.

Because every event in E corresponds to no more and no less than one event in T,



(meant to read: a function t is the mapping of the set E to the set T such that for all t's in the set T there exists one and only one e in E such that t as a function of e is equal to t)

such that for a bijector J between T and E,



so that



and



that is, the cardinality of both sets is equal such that the magnitude of both sets is equal:

.


---


Thank you for any and all input. Any and all critiques and edits are welcome.

P. S. Yes, starting out with the most fundamental definition of time is a goal, then to be worked out so as to be limited to the specified boundary condition which is so-bounded because of the set of E being the only set where T is relevant.

#8 2013-10-26 08:14:50

Reuel
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Re: Set notation for corresponding sets

As there are no replies I will assume there are no outrageous errors, except I think maybe I should have written



with the factorial if I wanted to say "there exists one and only one".

Thanks for the help, guys.

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