Set Theory Question

[Johnny5]

I have read probably over 100 different books on set theory over the past 15 years.

 

I have a question, it's about notation and that's all it is about.

 

Suppose that someone uses the following notation for a set:

 

{a,b,c}

 

Must I infer that the set has three elements in it OR

 

have they left open the possibility that the set contains one element, or possibly even two elements?

 

I could adjust my logical structure to accomodate anything, but the question is only about standard usage of the notation above.

 

Thank you

[Dave]

Erm... not that I'm aware of. Everything I have seen or read has always infered that if A = {a,b,c} then |A| = 3.

[Johnny5]

Erm... not that I'm aware of. Everything I have seen or read has always infered that if A = {a,b,c} then |A| = 3.

 

Ok, thanks. That makes life easier from a reasoning standpoint of course, it's just I am writing something, and I was covering this issue.

 

It's a reasoning issue actually. if i took the time to explain it to you you would follow, but i just needed the question answered.

 

Well perhaps for your own edification...

 

The title of the work is

 

"Treatise On The Transmission Of Knowledge"

 

Suppose that someone doesnt know anything about set theory at all.

 

They then see the following symbolism being used by the author, to explain set theory...

 

 

{a,b,c,d}

 

It is natural for them to initially wonder about several things...

 

One of which is, "does order matter"

 

Symbolically if order mattered, it would have been handled this way,

(a,b,c,d), by some particular author.

 

 

But see they don't know anything about that notation yet.

 

Now, assume they are accustomed to using the = symbol, and know the reflexive, transitive, and symmetric properties of equality.

 

They are well accustomed to having to draw the conclusion that such and such = something symbolically different, yet equivalent in some sense.

 

So they don't, a priori know if they must also do this with the set theoretic notation which uses the curly brackets.

 

Hence, if when that notation is used by some author, it is impossible for different symbols to denote the same element of the set, that author must supply the learner with that information, otherwise be remiss.

 

If you don't get it by now, don't worry about it.

 

You told me what I wanted to know.

 

I googled on "roster method" but got no clear answer to something that should be stated immediately in anyone's logical presentation of set theory.

 

Kind regards

 

PS: One more thing.

 

I am covering something called "tasking"

 

which has to do with demands made upon external reasoning agents, mental tasks that they must perform, in order to learn something from you.

 

When teaching, you seek to minimize the tasks they must carry out, while maximizing the rate of knowledge transfer.

 

but then this kind of thing is being covered in my treatise.

 

 

Also Dave, forgive me if I overanalyze. It happens by accident.

[Dave]

The one thing that I perhaps should have stated (skipped my mind earlier) is that

 

A = {a,b,c} => |A| = 3 if a, b and c are all distinct elements.

[Johnny5]

The one thing that I perhaps should have stated (skipped my mind earlier) is that

 

A = {a' date='b,c} => |A| = 3 if a, b and c are all distinct elements.[/quote']

 

Well see I don't want the 'if' left open.

 

I need notation which is unambiguous, and that is the point.

 

If that notation leaves open the possibility that some of the symbols being looked at inside the curly braces could denote the same element of the set, then I need a new notation which clearly says to the reasoning agent, there are 7 elements in this set... immediately, and you do not need to reason otherwise about it.

 

I was toying with this:

 

{a;b;c;d;e;f;g}

 

 

So if an author wants to leave the possibility open, as to what the symbols inside the braces denote, they can use

 

{a,b,c}

 

and if the author wants to close off the possibility, he can use the semicolon instead of the comma.

 

Obviously, this would only make sense in my work, but that's all its supposed to do.

 

But of course, the whole point of my question was this...

 

If it is already standard practice that when you see {x,y,z,p,d,q} that you can instantly infer (just by looking) that there are six elements in the set, then i needn't bother with the semicolon thing.

[Dave]

Unfortunately, I don't think there's a way around it. However, I'm probably not the best one to ask - not done this for a while

[matt grime]

Do not worry every one, for if the person who didn't know about sets was reading the definition for the first time and had taken the time to read it all properly they'd Know that sets do not have repeated elements and that they are unordered by definition. This whole nonsense about trying to infer something from notation alone is silly - why should they even infer that {a,b,c} is a set?

[Johnny5]

Do not worry every one, for if the person who didn't know about sets was reading the definition for the first time and had taken the time to read it all properly they'd Know that sets do not have repeated elements and that they are unordered by definition. This whole nonsense about trying to infer something from notation alone is silly - why should they even infer that {a,b,c} is a set?

 

I actually thought about this just this morning, and decided to do the following in my own work...

 

Earlier in this thread i said i was toying with the idea that using ; might help the reader infer that all the symbols inside the curly braces denote different elements of the set.

 

I decided to dispense with that idea completely.

 

The reason being that {a,b,c,d} is so widely used.

 

My point remains though. You cannot just pretend there was no point. The issue arose, because I am writing something, and I just want to be clear in my presentation of set theory, to a reader totally unfamiliar to it.

 

And exactly as Matt says, why should a reader even look at {a,b,c} and infer that the symbols denote a set. He should not. It has to be stated explicitely.

 

The thing is, I want to write something which is easy to understand, on your first reading.

 

Now, the most important thing about a set, is how many elements are in it.

 

So quibbling about whether or not {a,b,c}={a,b} because c=a is not a complete non-issue.

 

It is just as important to explain this kind of thing to the reader, as it is to explain to the reader that...

 

 

{a,b,c} denotes a set. (As Matt pointed out)

 

 

 

So here is what I finally settled on...

 

I will write using the following approach...

 

"In any discussion in which {a,b} denotes a set, unless explicitely stated otherwise, the symbols a,b denote different elements of the set."

 

 

The italicized portion is all that is required to avoid confusion, teach, and use currently accepted notation in the way it was intended.

 

If anyone has any helpful comments, they are welcome.

 

Regards

[matt grime]

The only comment I have is if this notional reader knew the proper definitions of a set then they'd not need to have that extra assumption explicitly italicized. There ought to be no confusion, and indeed there isn't one. You can't just pick up a maths book and expect to understand the material immediately if you do not know what the notation is.

[Johnny5]

The only comment I have is if this notional reader knew the proper definitions of a set

 

And what definition is that?

 

IN some texts 'set' is undefined.

 

In others, there is a somewhat goofy definition.

[ElijahJones]

Yup, however I can have a dataset {1,2,4,5,2,3,1}, but this is tricky because it is actually {o1,o2,o3,o4,o5,o6,o7} seven distinct osbervations and not necessarily ordered either. For the purpose of data analysis I do not want to do this {1,2,4,5,2,3,1}={1,2,3,4,5}. The means are 9 and 7.5 respectively. So context can sometimes make or break the intepretation of symbols, thought it should not be. I suppose we could use square bracecs for datasets ot some other such thing like {d:1,2,4,5,2,3,1}. Context is usually sufficient. But I will say that if you have read hundreds of books on set theory over fifteen years and you have never seen the definition Matt mentioned, umm you might want to change the way you read math texts.

 

EJ

[matt grime]

if you go around resurrectung very dead threads you might not get much response, especially since Johnny5 is now permanently banned from theis site.

[ElijahJones]

Hi Matt,

 

I'm new and looking at dates is not a habit yet.

 

Thanks

 

EJ