Set theory question

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bbq
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Set theory question

Postby bbq » Tue Mar 29, 2011 2:52 pm UTC

Say, if using set theory, I had a finite set called A with a range of values from 1 to n, arranged from smallest to greatest. How would I go about writing down the 'the 1st/2nd/3rd value from set A' without specifically defining what these values would be in the notation? Is this possible?
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jestingrabbit
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Re: Set theory question

Postby jestingrabbit » Tue Mar 29, 2011 3:03 pm UTC

bbq wrote:Say, if using set theory, I had a finite set called A with a range of values from 1 to n, arranged from smallest to greatest. How would I go about writing down the 'the 1st/2nd/3rd value from set A' without specifically defining what these values would be in the notation? Is this possible?


Normally you do something like "let [imath]A = \{a_i :\ 1\leq i\leq M,\ a_i< a_{i+1} \} \subseteq \{1\ldots n\}[/imath]" or maybe "let [imath]A=\{a_1, a_2, \ldots , a_m\}[/imath] such that [imath]1\leq a_i< a_{i+1} \leq n.[/imath]"
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Re: Set theory question

Postby mr-mitch » Tue Mar 29, 2011 3:28 pm UTC

You could always use a tuple as a nested set

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Yakk
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Re: Set theory question

Postby Yakk » Tue Mar 29, 2011 3:33 pm UTC

bbq wrote:Say, if using set theory, I had a finite set called A with a range of values from 1 to n, arranged from smallest to greatest.

In most set theory, sets are not "arranged" in an order.

If you want an arrangement or an order, you instead build a set of tuples, with an indexing element followed by the element in question.
How would I go about writing down the 'the 1st/2nd/3rd value from set A' without specifically defining what these values would be in the notation? Is this possible?

In that case, you'll note that it is really easy.

Now note that Jesting's clause indexed the elements -- a_i can be considered shorthand for the (partial) function a:N->X defined by a(i):=a_i.

Once you have done that, you can extract the ith element by referring to the element a_i.
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bbq
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Re: Set theory question

Postby bbq » Tue Mar 29, 2011 9:49 pm UTC

Yeah, thats what I'm looking for. Thanks guys.
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