(SR) Glossary of Set Theoretic Symbols

Good point D H. Using the AMS LaTex guide I tried \superseteq for superset, but it was denied. Glad to see you found it! I agree they are different concepts; let me straight away make amends, but note this thread was originally intended as a glossary of terms, not a set theory tutorial.

Ah, looks like I can't edit the opening post. Damn!

Anyway, I'm going to the pub, so I guess this is the last sense you'll get out of me today!

 

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Quark---

tell me what you want to change in your OP and I will edit it.

 

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I am not disputing any of the facts, but that rule you just stated is arbitray also. I.e., except for it, why is not a\(\cup \{b,c}\\) = {a,{b,c}} (Hope my stolen copy and modified of La tex worked.) By second edit: I now almost have it the { } around the b,c are strangely large I.e. I wanted something like: aU{b,c} = {a,{b,c}}.

Ain't "trial and error" grand?

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Yes I agree with your logic / proof, but if "union" were generalized and "mixed sets" were not also arbitarily excluded from being sets, I do not see any problem with the above result of a union between the element a and the set {b,c} being (mixed) set {a,{b,c}} which is what I hoped my Latex above would also state.

BTW, hope you enjoyed vist to pub. I will be away from Sao Paulo for 3 days beginning in a few hours, but may have internet access. I have not "given up" yet, but think I will stop trying to learn Latex simultaneously.

 

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OK, folks, I have three problems here. First is, I went to the pub (obvious consequence!)

Second, I sat on my spectacles, so I am really struggling here

Third, and most important; this thread is supposed to be about notation. If anyone wants to start a "set theory" tutorial, then please do. I had not intended this thread to be a tutorial in that sense.

 

Sorry I came across as grumpy last night - I was pissed off about my specs. Anyone is, of course free to post any set theory -related comment, ignore me.

Anyway, let me say a word or two about usage, as D H raised an interesting point.

If I write \(x \in X\) I mean something like this: given the set \(X\), then \(x\) is an element in \(X\). No harm is done if I use the negation:given the set \(X\) there are no elements in \(X\) - I have simply defined the empty set.

Now if I write \(X \ni x\) I mean something like: given the element \(x\), then \(X\) is a set containing \(x\). The negation - given the element \(x\) there is no set containing \(x\) is the Russell paradox, since this statement merely defines the set whose elements are members of no set.

Now consider \(A \subset B\). (the following is easier to see for proper subsets). Here I mean that all elements in \(A\) are also elements in \(B\), but there are elements in \(B\) that are not also elements in \(A\).

But here I am under an obligation to give rather specific instructions on how to find the subset. This is not usually especially onerous - all I require is a "such that" or "for which" statement. So I am saying : \(B\) is the set of which \( A\) is a subset.

Note there is no particular consequence to the negation - \(B\) may well have no non-empty proper subsets.

If I write \(B \supset A\), I am saying that given the set \( A\) then \(B\) is a superset of \(A\). But note, the negation does not imply a paradox, merely that \(A\) is not a subset of any other set.

In short; the assertion \(X \ni x\) must be true, the assertion \( X \supset Y\) need not be; they have, as assertions, a different status.

 

Sorry, in my eagerness to "sit on" Billy T, I missed these points:

Ben can you do this us please? (The placement in the list should be fairly obvious)

By the way, for proper subsets I much prefer \(X \subsetneq Y\), but as you see it doesn't render here.

You have the better of me here! By naive set theory do you mean a construction like 0 = {}, 1 = {{}} and so on? If so, it looks like you are right about "mixed sets". But I only know ZF, and even then, only what I need to know.

 

I have to say, that I have never encountered the usage practice you present here, QuarkHead. In my experience, the semantic content of \(X \ni x\) and \(x \in X\) is exactly the same: Set \(X\) contains the element \(x\) is equivalent to stating that element \(x\) is contained in set \(X\). The same for the negation: As far as I know, \(x \notin X\) and \(X \not \ni x\) mean exactly the same thing.

 

No, in naïve set theory one would not worry about the representation of the integers or anything like that, but merely assume that "they are there", so to speak. And the problems of considering pretty much anything as being a set, and the consequent problems is, of course, what led to the more rigourous approaches such as ZF(C).

I merely meant that in order to learn the concepts presented here, the rigour used in axiomatic set theory is not necessary, and there's no problem with considering, say, \(\{a,\{a\}\}\) as a set, even though it is "mixed".

 

You perhaps know better than I do, as, like most, I absorbed set theory more-or-less by osmosis. See if you can agree with this:

Let \(x\) be an object in the universe \(\mathcal{U}_o\) of objects. I will say that some set \(X\) is well-defined iff \(x \in X\) or \(x \notin X\).

Suppose now that \(X\) is well defined and that \(\forall x \in \mathcal{U}_o,\; x \notin X\), then \( X = \emptyset\).Right?

now suppose there is some set \( X\) in the universe \(\mathcal{U}_S\) of sets. If it is the case that there is some \(x \in \mathcal{U}_o\) where \( \forall X \in \mathcal{U}_S,\; X \not\ni x\) then this invites the wrath of Russell.

Hence my contention that \( x \notin X\) and \( X \not \ni x\) are logically inequivalent. No?

 

There is no absolute "universe" since it will lead to Russel's paradox. So we just construct sets starting from smaller ones. And \(x\in X\) and \(X\ni x\) are logically equivalent. They are only different in that when other symbols are present using one or the other you can make expressions shorter. For example, \(\exists X\ni x\) can be written as \(\exists X\) such that \(x\in X\).

 

What kind of backwards math are you doing?

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I've never seen statements written like this. I think a few of my professors would mark me off for using "ambiguous language" if I said \(\exists X\ni x\).

Of course, there's nothing wrong with it.

 

Yes it would had I chosen to call \(\mathcal{U}\) the universal set, but I didn't; \(\mathcal{U}\) was intended merely to denote " all objects/sets you you possibly imagine".

And at what point in this process do you feel Bertie's breath on your neck?

temur, you miss the point; suppose I write, given the set \(X\), that \( x \notin X\; \forall x\). How would you describe \(X\)? \(\emptyset\) perhaps?

Suppose I now write, given the point \(x\) that \( X \not \ni x\; \forall X\). What would you say to that? Russell paradox, maybe?

This, it seems to me, is a logical inequivalence

 

Both statements are equivalent (unless you think you can find a set \(X\) and and element \(x\) such that \(x \in X\) and \(X \not\ni x\), or vice-versa). In your definition of \(x \in X\) you've chosen to put emphasis on the fact that \(x\) is generally not the only element contained in \(X\), and for \(X \ni x\) you've stressed that \(X\) may not be the only set that contains \(x\), but both statements apply in general.

The negation of "\(x\) is an element of the set \(X\)" is "\(x\) is not an element of the set \(X\)".

"There are no elements in \(X\)" is the negation of "There exists at least one element in \(X\)".

The negation of "\(X\) is a set containing the element \(x\)" is "\(X\) is not a set containing the element \(x\)".

"There is no set which contains \(x\)" is the negation of "There exists at least one set which contains \(x\)".

1. How would this statement be different if you substituted \(X \not\ni x\) for \(x \not\in X\)?
2. I can define \(S_x := \{ x \}\) for any given \(x \in \mathcal{U}_o\), so no \(x\) will ever satisfy the condition above.

How about:

\(X\) such that \(X \not\ni x\; \forall x\)​

and

\(x\) such that \(x \not\in X\; \forall X\)​

?

 

I agree this is a bit strange, but I often saw professors writing like this. For example, I think \(X\ni x\mapsto y\in Y\) is standard.

edit: \mapsto shows a different symbol (\rightarrow) here.

 

Yes, I agree. It is often clear from the context what "universe" you are in. I think you are thinking in terms of operations and the order in which they are performed; but sometimes it is useful to think of mathematics just as symbols written on a piece of paper or blackboard. Actually there is no "time" in mathematics, things just exist.

 

http://en.wikipedia.org/wiki/Zermelo-Fraenkel_set_theory#Axioms

 

Well, you all seem intent on challanging a statement I never made!

Look. No sane person would deny that \( x \in X \Rightarrow X \ni x\), and I am sane (I think).

Likewise, clearly \( x \notin X \Rightarrow X \not \ni x\).

It's possible I am being unorthodox, and I accept that it is largely a matter of notational aesthetics. It is simply, as pryck noted, where one places the emphasis. But surely all would agree with the following:

If I write that, for some fixed set \(X\), that \(x \in X\), without an explicit quantifier, then I mean that \(x\) is a typical (generic) element in \(X\); anything I say about \(x\) can be taken to be true of all elements in \(X\), unless I say otherwise.

If I write that, for some fixed element \(x\), \(X \ni x\), it is not the case that whatever I say about \(X\) can be taken to be true of all sets containing the element \(x\).

All this is so, obviously, because \(X\) is a set, and \(x\) is an element.

As I say, it's merely notational aesthetics, and therefore scarcely something to be dogmatic about. Time to move on, I say!

 

Let's move on!

 

Excellent work... yet again... this place does indeed surprise me sometimes. :bawl:

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Now... [please] leave him alone. His work is mathematical perfect.