expressing finite amount of sets with notation

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expressing finite amount of sets with notation

Postby >-) » Fri Feb 14, 2014 3:56 pm UTC

i want to use those fancy math symbols to say that all intersections of any finite group of sets in T are in T.

If i had just two sets i would write something like

∀A∀B(A∩B⊃T)

how would i extend this to a finite amount of sets?

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Qaanol
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Re: expressing finite amount of sets with notation

Postby Qaanol » Fri Feb 14, 2014 4:33 pm UTC

“T is closed under finite intersections.”

Also, your words and your symbols do not say the same things for the n=2 case. Which provides a perfect segue for me to mention that words are much clearer than mathematical symbols in almost every case. The only advantage of symbols is brevity, and you should always explain in words what you are doing with symbols, and why.
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WibblyWobbly
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Re: expressing finite amount of sets with notation

Postby WibblyWobbly » Fri Feb 14, 2014 4:45 pm UTC

Qaanol wrote:“T is closed under finite intersections.”

Also, your words and your symbols do not say the same things for the n=2 case. Which provides a perfect segue for me to mention that words are much clearer than mathematical symbols in almost every case. The only advantage of symbols is brevity, and you should always explain in words what you are doing with symbols, and why.


The brevity of that first statement is a thing of beauty.

What I originally thought is to use something like "For all sets Ai such that Ai is a subset of T, the intersection of any finite set of Ai is contained in T ... " and then I realized that I was probably going in the opposite direction of a useful or perhaps even correct answer. And Qaanol's aside is very well-stated. Basically, always try to use the smallest number of words necessary, but always use all the necessary words. Symbols can omit necessary words.

Tirian
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Re: expressing finite amount of sets with notation

Postby Tirian » Fri Feb 14, 2014 4:59 pm UTC

In fact, you could say ∀A∈T∀B∈T(A∩B∈T), the fact that this extends to all finite intersections is a simple proof by induction.

In my mathematical training, I'd have said

∀𝒞 ∈Fin(T): ∩𝒞∈T

But there was an implicit agreement that Fin(S) = {T ∈ 𝒫(S)| T is finite}. If there is a universal notation for the finite power set, I don't know of it.

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Re: expressing finite amount of sets with notation

Postby lalop » Fri Feb 14, 2014 5:24 pm UTC

T_1,...,T_n ∈ T ⇒ T_1 ∩ ... ∩ T_n ∈ T

This is a little bit of an abuse of notation, in that it doesn't specify whether n is allowed to be 1.5, or what happens if n is 0. So if you need rigor, spell it out.

>-) wrote:∀A∀B(A∩B⊃T)

That actually reads "T is a subset of every intersection.. ever." Which implies that T is the empty set.

>-)
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Re: expressing finite amount of sets with notation

Postby >-) » Fri Feb 14, 2014 6:15 pm UTC

i see the mistake. i guess i'll go with words then, thanks!

Farabor
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Re: expressing finite amount of sets with notation

Postby Farabor » Sun Feb 16, 2014 7:36 am UTC

If you really wanted to do this, this should work:
[imath]∀A,∃nN #A=n⇒(AT⇒∩aA aT)[/imath]

This'd read, unless sleep dep is messing with me: For all nonempty subsets of T with finite cardinality, the intersection of all of the elements of that subset is also in T, which is I believe what you were asking for.

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Re: expressing finite amount of sets with notation

Postby lorb » Sun Feb 16, 2014 11:25 am UTC

Qaanol wrote:“T is closed under finite intersections.”

Also, your words and your symbols do not say the same things for the n=2 case. Which provides a perfect segue for me to mention that words are much clearer than mathematical symbols in almost every case. The only advantage of symbols is brevity, and you should always explain in words what you are doing with symbols, and why.


A little nitpick: A word that has been defined by the mathematical community to represent a specific concept/idea is no less a mathematical symbol than ∀.
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