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Math Forum / Mathematics / Algebra Help / July 2009


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Writing a set definition

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Axis - 29 Jun 2009 17:27 GMT
Let J be any set of primes. For any integer i, let Div(J, i) = {m in J: m |
i}.
'|P A' shall denote 'the set of sets A'.

I'm writing a definition that I've currently put as follows:
|P{Div(J, i): Div(J\{r}, i) > 0, r | i }: i in [x,y].

Should I have curly braces around everything except the |P, to make it
|P{{Div(J, i): Div(J\{r}, i) > 0, r | i }: i in [x,y]}?

Am I right in saying that if I had put
{Div(J, i): Div(J\{r}, i) > 0, r | i, i in [x,y] }
I would have union of the set I have currently written?

While I'm at it, I want to refer once more to something Paul mentioned,
which is that there is never any excuse for using a calculation as a dummy
variable in a definition. My question is, does a cardinality, such as
|[x,y]|, count as a calculation?

WTIA.
Reply to this Message
Axis - 29 Jun 2009 17:55 GMT
PS
<<Should I have curly braces around everything except the |P, to make it
|P{{Div(J, i): Div(J\{r}, i) > 0, r | i }: i in [x,y]}?>>

My object in forming the set of sets is to make a set of the elements Div(J,
i) for i ranging over [x,y].
I should also correct the set element to Div(J\{r}, i) ('\' being Boolean
subtraction), so I've now got
"|P{{Div(J\{r}, i): Div(J\{r}, i) > 0, r | i }: i in [x,y]}".

> Let J be any set of primes. For any integer i, let Div(J, i) = {m in J: m
> | i}.
[quoted text clipped - 16 lines]
>
> WTIA.
Reply to this Message
Frederick Williams - 29 Jun 2009 19:22 GMT
> Let J be any set of primes. For any integer i, let Div(J, i) = {m in J: m |
> i}.
> '|P A' shall denote 'the set of sets A'.

Do you mean 'the set of subsets of A'?

> I'm writing a definition that I've currently put as follows:
> |P{Div(J, i): Div(J\{r}, i) > 0, r | i }: i in [x,y].
>
> Should I have curly braces around everything except the |P, to make it
> |P{{Div(J, i): Div(J\{r}, i) > 0, r | i }: i in [x,y]}?

The second one.

> Am I right in saying that if I had put
> {Div(J, i): Div(J\{r}, i) > 0, r | i, i in [x,y] }
[quoted text clipped - 6 lines]
>
> WTIA.

Signature

Which of the seven heavens / Was responsible her smile /
Wouldn't be sure but attested / That, whoever it was, a god /
Worth kneeling-to for a while / Had tabernacled and rested.

Reply to this Message
Axis - 29 Jun 2009 21:24 GMT
>> Let J be any set of primes. For any integer i, let Div(J, i) = {m in J: m
>> |
>> i}.
>> '|P A' shall denote 'the set of sets A'.
>
> Do you mean 'the set of subsets of A'?

No, the set of sets A where A is defined as satisfying < such-and-such >. I
obviously missed out some detail.

>> I'm writing a definition that I've currently put as follows:
>> |P{Div(J, i): Div(J\{r}, i) > 0, r | i }: i in [x,y].
[quoted text clipped - 3 lines]
>
> The second one.

The doubled left braces are OK then...?
Reply to this Message
Frederick Williams - 30 Jun 2009 10:17 GMT
> >> Let J be any set of primes. For any integer i, let Div(J, i) = {m in J: m
> >> |
[quoted text clipped - 5 lines]
> No, the set of sets A where A is defined as satisfying < such-and-such >. I
> obviously missed out some detail.

I don't know what you mean.  Let's suppose, for the sake of
definiteness, that A satisfies the condition: 'having elements a and b
and no others'.  So A = {a, b}, now what is 'the set of sets A'?

> >> I'm writing a definition that I've currently put as follows:
> >> |P{Div(J, i): Div(J\{r}, i) > 0, r | i }: i in [x,y].
[quoted text clipped - 5 lines]
>
> The doubled left braces are OK then...?

Signature

Which of the seven heavens / Was responsible her smile /
Wouldn't be sure but attested / That, whoever it was, a god /
Worth kneeling-to for a while / Had tabernacled and rested.

Reply to this Message
Axis - 30 Jun 2009 11:44 GMT
>> >> Let J be any set of primes. For any integer i, let Div(J, i) = {m in
>> >> J: m
[quoted text clipped - 11 lines]
> definiteness, that A satisfies the condition: 'having elements a and b
> and no others'.  So A = {a, b}, now what is 'the set of sets A'?

Let's say A satisfies the condition of having in it at least one element, r
in an interval [x,y], that is divisible by a prime p in a set of primes P.
So A = {r in [x,y]: p | r): p in P.
Or should I say A(x,y,p) = {r in [x,y]: p |r} p in P.
My set of sets would have every p in P as being a divisor of the set
element.
The set I am formulating is more complex, and I want to end up with a set of
sets of integers, not a set of integers.
Cheers.
Reply to this Message
Frederick Williams - 30 Jun 2009 14:33 GMT
> >> >> Let J be any set of primes. For any integer i, let Div(J, i) = {m in
> >> >> J: m
[quoted text clipped - 15 lines]
> in an interval [x,y], that is divisible by a prime p in a set of primes P.
> So A = {r in [x,y]: p | r): p in P.

I think you mean A = {r in [x,y]: p | r, p in P}.

> Or should I say A(x,y,p) = {r in [x,y]: p |r} p in P.
> My set of sets would have every p in P as being a divisor of the set
> element.
> The set I am formulating is more complex, and I want to end up with a set of
> sets of integers, not a set of integers.
> Cheers.

Signature

Which of the seven heavens / Was responsible her smile /
Wouldn't be sure but attested / That, whoever it was, a god /
Worth kneeling-to for a while / Had tabernacled and rested.

Reply to this Message
Axis - 30 Jun 2009 14:39 GMT
>> >> >> Let J be any set of primes. For any integer i, let Div(J, i) = {m
>> >> >> in
[quoted text clipped - 21 lines]
>
> I think you mean A = {r in [x,y]: p | r, p in P}.

Well the idea is to illustrate the problem, which is that I want a set S of
sets M, and each set M is a set of multiples, in [x,y], of a given divisor.
That's not exactly what I have in the work I'm doing, but it's a simplified
version of it.
Cheers.
Reply to this Message
Frederick Williams - 30 Jun 2009 15:11 GMT
> >> >> >> Let J be any set of primes. For any integer i, let Div(J, i) = {m
> >> >> >> in
[quoted text clipped - 27 lines]
> version of it.
> Cheers.

So let's call A, A(x,y,P) then

 {A(x,y,P): conditions on x, y and P}

is a set of sets A.

Note: not A(x,y,p) with a small p, and don't use |P.

Signature

Which of the seven heavens / Was responsible her smile /
Wouldn't be sure but attested / That, whoever it was, a god /
Worth kneeling-to for a while / Had tabernacled and rested.

Reply to this Message
Paul Sperry - 30 Jun 2009 05:58 GMT
> Let J be any set of primes. For any integer i, let Div(J, i) = {m in J: m |
> i}.
> '|P A' shall denote 'the set of sets A'.

A really bad idea.

> I'm writing a definition that I've currently put as follows:
> |P{Div(J, i): Div(J\{r}, i) > 0, r | i }: i in [x,y].
[quoted text clipped - 5 lines]
> {Div(J, i): Div(J\{r}, i) > 0, r | i, i in [x,y] }
> I would have union of the set I have currently written?

No

You write elsewhere in this thread:
  "My object in forming the set of sets is to make a set of the
  elements Div(J, i) for i ranging over [x,y]."

Do you mean Div(J, [x, y])?
Or do you mean { Div(J, i) : i in [x, y] }?
Or something else?

You would be doing yourself a _big_ favor if you would *stop* trying to
use fancy notation in which you are untrained and simply use clear,
concise, unambiguous English. You would be doing your readers a favor
too.

I'll go even further; unless you can express your ideas in precise and
understandable English, you have no hope of correctly using Math
shorthand.

It seems that most beginners are unwilling to try out their ideas on an
example or two. You certainly seem to fall into that category.

I have pretty much stopped replying to your posts because I am
unwilling to take the time or make the effort needed to untangle what I
suppose are your attempts to seem to be a "pro".

> While I'm at it, I want to refer once more to something Paul mentioned,
> which is that there is never any excuse for using a calculation as a dummy
> variable in a definition. My question is, does a cardinality, such as
> |[x,y]|, count as a calculation?

Yes. A variable is a variable - it is, informally and ungrammatically,
something which may be substituted for.

Signature

Paul Sperry
Columbia, SC (USA)

Reply to this Message
Axis - 30 Jun 2009 12:00 GMT
>> Let J be any set of primes. For any integer i, let Div(J, i) = {m in J: m
>> |
>> i}.
>> '|P A' shall denote 'the set of sets A'.
>
> A really bad idea.

I am actually using mathcal{P}, which I was told was fairly standard. That's
OK isn't it?

>> I'm writing a definition that I've currently put as follows:
>> |P{Div(J, i): Div(J\{r}, i) > 0, r | i }: i in [x,y].
[quoted text clipped - 20 lines]
> concise, unambiguous English. You would be doing your readers a favor
> too.

The 'fancy notation' was suggested to me by members of this NG. After
sticking with my old notation for a while, to the disgruntlement of those
who suggested it, I started using it and haven't turned back. I didn't know
you had to be trained in notation to use it.
>> While I'm at it, I want to refer once more to something Paul mentioned,
>> which is that there is never any excuse for using a calculation as a
[quoted text clipped - 4 lines]
> Yes. A variable is a variable - it is, informally and ungrammatically,
> something which may be substituted for.

So, in defining a set, what do I write when I have an interval [x,y] of
integers and, when using the set in an exposition, I want to indicate
dependency upon |{x,y]|?

To give an example, I have a set of intervals [x+i. y+i], of length y-x+1,
for all of which sum{|Div(J, i): i in [x+i, y+i]}: i in |N is equal. Now I
want some other intervals of a different length but which share this
property. And I want to end up with a definition of a set of sets that has
this dependency on interval length.

With thanks.
Reply to this Message
Axis - 30 Jun 2009 14:45 GMT
>>> While I'm at it, I want to refer once more to something Paul mentioned,
>>> which is that there is never any excuse for using a calculation as a
[quoted text clipped - 4 lines]
>> Yes. A variable is a variable - it is, informally and ungrammatically,
>> something which may be substituted for.

Another matter: if you object (and I am not entirely sure, from what you
have said, that you *do* object) to the use of cardinality as an argument
'a', 'b' or 'c' in H(a,b,c), why did you not say so in my previous thread?
In particular, I am thinking of this comment of yours made in a post of 28th
June:
<<x and y are bound variables in the expresson
Y_L = { [x,y] subset R | y - x = L }>>

(The cardinality of [x,y], of course, is y-x+1, which obfuscates it a little
further...).
Cheers.
Reply to this Message
Paul Sperry - 01 Jul 2009 03:38 GMT
> >>> While I'm at it, I want to refer once more to something Paul mentioned,
> >>> which is that there is never any excuse for using a calculation as a
[quoted text clipped - 12 lines]
> <<x and y are bound variables in the expresson
> Y_L = { [x,y] subset R | y - x = L }>>

_I_ didn't write that.

> (The cardinality of [x,y], of course, is y-x+1, which obfuscates it a little
> further...).
> Cheers.

Signature

Paul Sperry
Columbia, SC (USA)

Reply to this Message
Brian M. Scott - 01 Jul 2009 05:27 GMT
On Tue, 30 Jun 2009 22:38:43 -0400, Paul Sperry
<plsperry@sc.rr.com> wrote in
<news:300620092238437272%plsperry@sc.rr.com> in
alt.algebra.help:

[...]

>> In particular, I am thinking of this comment of yours
>> made in a post of 28th  June:

>> <<x and y are bound variables in the expresson
>> Y_L = { [x,y] subset R | y - x = L }>>

> _I_ didn't write that.

That was William Elliot.

[...]

Brian
Reply to this Message
Paul Sperry - 01 Jul 2009 03:37 GMT
> >> Let J be any set of primes. For any integer i, let Div(J, i) = {m in J: m
> >> |
[quoted text clipped - 5 lines]
> I am actually using mathcal{P}, which I was told was fairly standard. That's
> OK isn't it?

No. Some variant of P(A) is often used to denote the power set of a set
A. The power set, P(A), is the set of all subsets of set A. That is a
long way from what you wrote.

[...]

> > You would be doing yourself a _big_ favor if you would *stop* trying to
> > use fancy notation in which you are untrained and simply use clear,
[quoted text clipped - 5 lines]
> who suggested it, I started using it and haven't turned back. I didn't know
> you had to be trained in notation to use it.

And yet almost all of your posts are questions about notation.

> >> While I'm at it, I want to refer once more to something Paul mentioned,
> >> which is that there is never any excuse for using a calculation as a
[quoted text clipped - 14 lines]
> property. And I want to end up with a definition of a set of sets that has
> this dependency on interval length.

An excellent example. What in the world are you trying to say? What is
one to make of this? What has the "i" in [x + i, y + i] got to do with
anything? What set of intervals? In "sum{|Div(J, i): i in [x+i, y+i]}:
i in |N" you are using "i" in three different ways. What about "i in
|N"? That's an infinite number of i's. There's more. (Never mind the
missing cardinal bar.)

Here's an exercise: Write this out as clearly and precisely as you can
- give it your best shot.

Signature

Paul Sperry
Columbia, SC (USA)

Reply to this Message
Axis - 01 Jul 2009 13:56 GMT
>> >> Let J be any set of primes. For any integer i, let Div(J, i) = {m in
>> >> J: m
[quoted text clipped - 11 lines]
> A. The power set, P(A), is the set of all subsets of set A. That is a
> long way from what you wrote.

Oh blimey -- that came from the tex users group; usually when one them is in
the slightest bit misinformed, all hell breaks loose.

Anyhow, before I continue with the 'cardinality/dummy variable' issue, the
matter of \mathcal{P} has reminded me of something. There are various other
symbols that I'm wondering about. Should any of the following get a mention
in my introduction?
ln(x) = natural log of x
\sum{A}, i.e. Sigma A = sum of the set of members of A
|A-B| = absolute difference between A and B (I have already said that |A|
sgnifies the cardinality of A; at places I've got '||A| - |B||')?

With thanks.
Reply to this Message
Frederick Williams - 01 Jul 2009 15:07 GMT
> |A-B| = absolute difference between A and B (I have already said that |A|
> sgnifies the cardinality of A; at places I've got '||A| - |B||')?

I'm confused.  Are A, B sets or numbers?

Signature

Which of the seven heavens / Was responsible her smile /
Wouldn't be sure but attested / That, whoever it was, a god /
Worth kneeling-to for a while / Had tabernacled and rested.

Reply to this Message
Axis - 01 Jul 2009 16:25 GMT
>> |A-B| = absolute difference between A and B (I have already said that |A|
>> sgnifies the cardinality of A; at places I've got '||A| - |B||')?
>
> I'm confused.  Are A, B sets or numbers?

Sets.
Reply to this Message
Frederick Williams - 02 Jul 2009 09:45 GMT
> >> |A-B| = absolute difference between A and B (I have already said that |A|
> >> sgnifies the cardinality of A; at places I've got '||A| - |B||')?
> >
> > I'm confused.  Are A, B sets or numbers?
>
> Sets.

In that case, |A - B| is the number of elements in the set whose
elements are the elements of the set A minus the elements of the set B.
Nobody (with possibly one exception) calls that the absolute difference
between A and B.  You can see, can't you, that

  |A - B|  doesn't necessarily equal  ||A - |B||

?
Signature

Which of the seven heavens / Was responsible her smile /
Wouldn't be sure but attested / That, whoever it was, a god /
Worth kneeling-to for a while / Had tabernacled and rested.

Reply to this Message
Axis - 02 Jul 2009 12:28 GMT
>> >> |A-B| = absolute difference between A and B (I have already said that
>> >> |A|
[quoted text clipped - 12 lines]
>
> ?

Sorry I was confusing the whole thing by using A and B to mean two different
things.
First, when I said |A-B| I meant them to be numbers, so instead I'll write
|i - j|; and then I also have ||A|-|B|| where A and B are sets. I have
already said in my introduction that |A| is to denote the cardinality of A.
I am just curious to know whether this is the absolute difference warrants a
similar mention.
Likewise for \sum = sum of...
and ln = natural logarithm.
Cheers.
Reply to this Message
Frederick Williams - 02 Jul 2009 13:36 GMT
> I am just curious to know whether this is the absolute difference warrants a
> similar mention.
> Likewise for \sum = sum of...
> and ln = natural logarithm.

None of these three needs defining.

Signature

Which of the seven heavens / Was responsible her smile /
Wouldn't be sure but attested / That, whoever it was, a god /
Worth kneeling-to for a while / Had tabernacled and rested.

Reply to this Message
Axis - 02 Jul 2009 15:05 GMT
> Here's an exercise: Write this out as clearly and precisely as you can
> - give it your best shot.

Here we go:

Let J be any set of primes. Let Y_L be the set of intervals of positive
integers, of length L. Let
S(L, s, J) = {I : |Mult(I, prod(J))| = s : I in Y_L.
My typical use of that set is in such a term as
"S(|I|, a, M) /\ S(|I|, b, N)".

For example, I have
Let U be the set of subsets of J.
For M,N in U, let E(I, a, b, M,N) be the set of {I_1, I_2} \subset S(|I|, a,
M) /\ S(|I|, b, N) such that
|{m in I_1: prod(M) | m or prod(N) | m}| < |{k in I_2: prod(M) | k or
prod(N) | k}|.

Any advice on the above set, what with its five arguments, would be most
welcome. I include those arguments for the sole reason that, as William
Elliot has informed me, the dependency (apparently) prescribes it.
I've got a bit of a different concern with that set, which is that if |M \/
N| = 1 I don't know whether it would be said to be undefined or empty. The
way I have put it is that {E(I, a, b, M,N): a, b in |N, M,N in U} is empty
if J=1. Is that OK?

I have also got, at a different place,
Let C(x,y,q) = \/({A'((x+y)/2, r): r in Q(n) \setminus {q}} \/ F(x,
(x+y)/2)) /\ A((x+y)/2,q)) /\ [0, (x+y)/2-1].

Providing you with the definitions of the sets on the RHS surely won't help
answer the question I have, which is whether I should replace (x+y)/2,
everywhere it is written in the above definition, with a dummy variable k.
If I do that, I'll have an extra argument, so it'll be C(x,y,k,q), which
looks excessive.

Incidentally, I am using X(x,y) as a set. I want it to stick in the mind,
which is why I am using 'X'. But is 'X' conventionally used to denote some
specific kind of variable?

With thanks.
Reply to this Message
Frederick Williams - 02 Jul 2009 21:08 GMT
> > Here's an exercise: Write this out as clearly and precisely as you can
> > - give it your best shot.
[quoted text clipped - 4 lines]
> integers, of length L. Let
> S(L, s, J) = {I : |Mult(I, prod(J))| = s : I in Y_L.

Where is {'s mate?

Signature

Which of the seven heavens / Was responsible her smile /
Wouldn't be sure but attested / That, whoever it was, a god /
Worth kneeling-to for a while / Had tabernacled and rested.

Reply to this Message
Axis - 02 Jul 2009 23:17 GMT
>> > Here's an exercise: Write this out as clearly and precisely as you can
>> > - give it your best shot.
[quoted text clipped - 6 lines]
>
> Where is {'s mate?

One of these '}' goes on the end.
Reply to this Message
Paul Sperry - 03 Jul 2009 04:47 GMT
> > Here's an exercise: Write this out as clearly and precisely as you can
> > - give it your best shot.
[quoted text clipped - 3 lines]
> Let J be any set of primes. Let Y_L be the set of intervals of positive
> integers, of length L.

By the way, the length of [x, y] is _not_ y - x + 1.

> Let
> S(L, s, J) = {I : |Mult(I, prod(J))| = s : I in Y_L.

"For positive integers k and s and finite set of primes J, let
S(k, s, J) be the set of all intervals [x, y] such that y -x = k and
exactly s elements of [x, y] are divisible by prod(J)."

So called "set builder" notation goes like this:
{ <context> <delimiter> <pass/fail condition> }
Eg: { n in Z : 2 | n }. "n in Z" is the context; ":" is the delimiter;
2 | n is the condition.

> My typical use of that set is in such a term as
> "S(|I|, a, M) /\ S(|I|, b, N)".
[quoted text clipped - 5 lines]
> |{m in I_1: prod(M) | m or prod(N) | m}| < |{k in I_2: prod(M) | k or
> prod(N) | k}|.

That makes no sense. Do you perhaps mean that for positive integers k,
a and b and subsets M and N of J, E(k, a, b, M, N) is the set of _all_
ordered pairs (I_1, I_2) of elements of S(k, a, M) /\ S(k, b, N) such
that there are fewer elements of I_1 divisible by either prod(M) or
prod(N) than there are similar elements of I_2?

[...]

If you would ever actually do the math, the notation would probably
take care of itself.

Signature

Paul Sperry
Columbia, SC (USA)

Reply to this Message
Axis - 03 Jul 2009 14:02 GMT
Paul,

>> Let J be any set of primes. Let Y_L be the set of intervals of positive
>> integers, of length L.
>
> By the way, the length of [x, y] is _not_ y - x + 1.

Eh? But I've been speaking of the 'length of an interval [x,y]', all along
in my paper, as being |[x,y]|! So is its length, then, y-x?

>> Let
>> S(L, s, J) = {I : |Mult(I, prod(J))| = s : I in Y_L.
[quoted text clipped - 7 lines]
> Eg: { n in Z : 2 | n }. "n in Z" is the context; ":" is the delimiter;
> 2 | n is the condition.

I did it OK, though, didn't I?

>> My typical use of that set is in such a term as
>> "S(|I|, a, M) /\ S(|I|, b, N)".
[quoted text clipped - 12 lines]
> that there are fewer elements of I_1 divisible by either prod(M) or
> prod(N) than there are similar elements of I_2?

Yes, but why would "the set of {I_1, I_2} \subset S(|I|, a,
> M) /\ S(|I|, b, N) such that..." be inadmissible? And how would I write
> your version formally?

What did you make of my query over the distinction between the set's being
empty and its being undefined, if |J| = 1?

With thanks.
Reply to this Message
Axis - 03 Jul 2009 22:03 GMT
I can only guess that the set I'm after is

For M,N in U [U being the set of subsets of J], let

E(I, a, b, M,N) = {{ I_1, I_2) : I_1, I_2 in S(|I|, a, M) /\ S(|I|, b, N),
|{m in I_1: prod(M) | m or prod(N) | m}| < |{k in I_2: prod(M) | k or
prod(N) | k}| }.

What about my
C(x,y,q) = \/({A'((x+y)/2, r): r in Q(n) \setminus {q}} \/ F(x,
(x+y)/2)) /\ A((x+y)/2,q)) /\ [0, (x+y)/2-1].

Shall I add the extra argument k and get rid of all the '(x+y)/2'?

WTIA.
Reply to this Message
Paul Sperry - 04 Jul 2009 03:54 GMT
> I can only guess that the set I'm after is
>
> For M,N in U [U being the set of subsets of J], let

Why not just say that M and N are subsets of J?

> E(I, a, b, M,N) = {{ I_1, I_2) : I_1, I_2 in S(|I|, a, M) /\ S(|I|, b, N),
> |{m in I_1: prod(M) | m or prod(N) | m}| < |{k in I_2: prod(M) | k or
> prod(N) | k}| }.

{ I_1, I_2) ?

I prefer the text version to the symbolic one - take your choice.

> What about my
> C(x,y,q) = \/({A'((x+y)/2, r): r in Q(n) \setminus {q}} \/ F(x,
> (x+y)/2)) /\ A((x+y)/2,q)) /\ [0, (x+y)/2-1].

What about it?

> Shall I add the extra argument k and get rid of all the '(x+y)/2'?

Without knowing how x and y figure in I can't possibly say.

Signature

Paul Sperry
Columbia, SC (USA)

Reply to this Message
Axis - 04 Jul 2009 13:16 GMT
Paul,

> { I_1, I_2) ?
>
> I prefer the text version to the symbolic one - take your choice.

I thought that was all done for usenet. I was about to change all my P(n)
and Q(n) to P_n and Q_n. I'll heed your advice, but surely for a prime, p_n
is more standard, and therefore better, than p(n)?

>> What about my
>> C(x,y,q) = \/({A'((x+y)/2, r): r in Q(n) \setminus {q}} \/ F(x,
[quoted text clipped - 5 lines]
>
> Without knowing how x and y figure in I can't possibly say.

As long as you don't object to the use of a calculation in a definition, I'm
happy. I had just got a bit wary when you had told me that there is no
excuse for using a calculation in place of a dummy variable in a definition.
Which brings me back to my original query, of whether using a cardinality,
(or a cardinality minus one, as in my 'L' for Y_L), counts as using a
calculation and is therefore not acceptable in a definition?

With thanks.
Reply to this Message
Paul Sperry - 04 Jul 2009 03:53 GMT
> Paul,
>
[quoted text clipped - 5 lines]
> Eh? But I've been speaking of the 'length of an interval [x,y]', all along
> in my paper, as being |[x,y]|! So is its length, then, y-x?

Take your ruler and examine the interval from 0 inches to 2 inches.
That interval is 3 inches long??

> >> Let
> >> S(L, s, J) = {I : |Mult(I, prod(J))| = s : I in Y_L.
[quoted text clipped - 9 lines]
>
> I did it OK, though, didn't I?

Did you match the template?

> >> My typical use of that set is in such a term as
> >> "S(|I|, a, M) /\ S(|I|, b, N)".
[quoted text clipped - 16 lines]
> > M) /\ S(|I|, b, N) such that..." be inadmissible? And how would I write
> > your version formally?

"The set of {I_1, I_2} .." is redundant. Worse, according to you, which
interval comes first makes a difference. Hence the need for ordered
pairs.

My version, although it could use a little editing, is as formal as it
needs to be; moreover, it does not suffer if it is spread over several
lines. The symbolic version won't fit on one line and it is not good to
have line breaks in such things.

> What did you make of my query over the distinction between the set's being
> empty and its being undefined, if |J| = 1?

Beats me. I've no idea what you're talking about.

Signature

Paul Sperry
Columbia, SC (USA)

Reply to this Message
Axis - 04 Jul 2009 13:25 GMT
Paul,

>> >> Let
>> >> S(L, s, J) = {I : |Mult(I, prod(J))| = s : I in Y_L.
[quoted text clipped - 11 lines]
>
> Did you match the template?

Oh - whoops - I suppose it should have been

S(L, s, J) = {I : |Mult(I, prod(J))| = s, I in Y_L}.

> "The set of {I_1, I_2} .." is redundant. Worse, according to you, which
> interval comes first makes a difference. Hence the need for ordered
> pairs.

One question about ordered pairs: is it admissible to have such a pair as
(a,a)? And if one says "M,N \subset J", or maybe "subsets M and N of J", is
that prescribing that M and N are distinct subsets of J? (Perhaps the word I
should use there is "unique", instead of "distinct"?).

A final matter; Is it OK to define my Y(L) (or 'Y_L', as it was), as 'any
set of intervals of length L', and then follow it with 'Choose L so that L >
prod(K)?" It might seem to contradict the use of 'any'; but it's not till a
little later on that I introduce K.

With thanks.
Reply to this Message
Axis - 04 Jul 2009 16:08 GMT
> Oh - whoops - I suppose it should have been
>
> S(L, s, J) = {I : |Mult(I, prod(J))| = s, I in Y_L}.

Correction:

S(L, s, J) = {I : |Mult(I, prod(J))| = s} :  I in Y_L.

But I'll go with your version.
Reply to this Message
Frederick Williams - 04 Jul 2009 19:01 GMT
> > Oh - whoops - I suppose it should have been
> >
[quoted text clipped - 5 lines]
>
> But I'll go with your version.

What does the second colon mean?

Since the first two I's are bound variables, the third one cannot denote
whatever they denote.

Signature

Which of the seven heavens / Was responsible her smile /
Wouldn't be sure but attested / That, whoever it was, a god /
Worth kneeling-to for a while / Had tabernacled and rested.

Reply to this Message
Axis - 04 Jul 2009 22:14 GMT
>> > Oh - whoops - I suppose it should have been
>> >
[quoted text clipped - 10 lines]
> Since the first two I's are bound variables, the third one cannot denote
> whatever they denote.

I thought I was saying, the element for the desired set is an interval I,
and the filter for these I's is that  |Mult(I, prod(J))| = s; and, finally,
any I in the set has a length L (ie. is in Y_L).
But maybe that last criterion limits me to a single I. So I guess it should
be on the other side of the right-hand-side brace, as I originally had it.
Cheers.
Reply to this Message
Frederick Williams - 05 Jul 2009 11:45 GMT
> >> > Oh - whoops - I suppose it should have been
> >> >
[quoted text clipped - 17 lines]
> be on the other side of the right-hand-side brace, as I originally had it.
> Cheers.

Yes

{x: condition 1 on x} : condition 2 on x

just isn't mathematical notation as I know it, but

{x: condition 1 on x and condition 2 on x}

is.

You could write

S(L, s, J) = {I in Y_L : |Mult(I, prod(J))| = s}.

Signature

Which of the seven heavens / Was responsible her smile /
Wouldn't be sure but attested / That, whoever it was, a god /
Worth kneeling-to for a while / Had tabernacled and rested.

Reply to this Message
Frederick Williams - 04 Jul 2009 19:14 GMT
> > Oh - whoops - I suppose it should have been
> >
[quoted text clipped - 5 lines]
>
> But I'll go with your version.

What does the second colon mean?

Since the first two I's are bound variables, your "correction" could
equally well be written

S(L, s, J) = {x : |Mult(x, prod(J))| = s} :  I in Y_L.

I cancelled an earlier vision of this because it was bollocks.

Signature

Which of the seven heavens / Was responsible her smile /
Wouldn't be sure but attested / That, whoever it was, a god /
Worth kneeling-to for a while / Had tabernacled and rested.

Reply to this Message
Axis - 04 Jul 2009 22:23 GMT
>> > Oh - whoops - I suppose it should have been
>> >
[quoted text clipped - 14 lines]
>
> I cancelled an earlier vision of this because it was bollocks.

I thought I understood your original objection. But this I definitely don't.
If I write "I" within curly braces, surely if I follow it with ": I ...", I
am, by the final, outside-the-braces condition, picking a single "I" to
which the other conditions are restricted?
Reply to this Message
Frederick Williams - 05 Jul 2009 11:40 GMT
> >> > Oh - whoops - I suppose it should have been
> >> >
[quoted text clipped - 19 lines]
> am, by the final, outside-the-braces condition, picking a single "I" to
> which the other conditions are restricted?

No, v in {v:...v...} is a bound variable and may be changed to any other
variable of the same type(*) so long as there is no clash.  Any v
outside {...} is a different v.

(* 'Type' is almost a technical term. )

Signature

Which of the seven heavens / Was responsible her smile /
Wouldn't be sure but attested / That, whoever it was, a god /
Worth kneeling-to for a while / Had tabernacled and rested.

Reply to this Message
Paul Sperry - 05 Jul 2009 05:26 GMT
> Paul,
>
[quoted text clipped - 17 lines]
>
>  S(L, s, J) = {I : |Mult(I, prod(J))| = s, I in Y_L}.

<context> = ?
<delimiter> = ?
<pass/fail condition> = ?

[...]

Signature

Paul Sperry
Columbia, SC (USA)

Reply to this Message
Axis - 05 Jul 2009 14:58 GMT
>> Paul,
>>
[quoted text clipped - 20 lines]
>
> <context> = ?

I

> <delimiter> = ?

> <pass/fail condition> = ?

|Mult(I, prod(J))| = s, I in Y_L

I am wondering what you objection is, if such you have.
Reply to this Message
Paul Sperry - 06 Jul 2009 05:13 GMT
> >> Paul,
> >>
[quoted text clipped - 22 lines]
>
> I

?? Look at my example again.

> > <delimiter> = ?
>
[quoted text clipped - 3 lines]
>
> I am wondering what you objection is, if such you have.

Signature

Paul Sperry
Columbia, SC (USA)

Reply to this Message
Axis - 06 Jul 2009 14:11 GMT
Paul,

>> >> >> >> Let
>> >> >> >> S(L, s, J) = {I : |Mult(I, prod(J))| = s : I in Y_L.
[quoted text clipped - 9 lines]
>> >> >> > delimiter;
>> >> >> > 2 | n is the condition.

I wonder about the strictness of the form the context should take.
A mathematician wrote a set definition for me that went as follows:

For q in Q(n), let A(k, q) = {i : 0 <= i <= k, q | k+i}
(if I recall correctly).
I thought I had picked up a subtle difference between "0 <= i <= k" and "i
in [0, k]", it being a matter of usage: the former is used for contexts in
which i ranges over the full interval [0, k], whereas the latter i is used
specifically to imply a fixed value. Perhaps I am mistaken....
Anyhow, would the following be synonymous with the above definition
"For q in Q(n), let A(k, q) = {i in [0, k] : q | k+i}" ?

With thanks.
Reply to this Message
Paul Sperry - 07 Jul 2009 05:41 GMT
> Paul,
>
[quoted text clipped - 17 lines]
> For q in Q(n), let A(k, q) = {i : 0 <= i <= k, q | k+i}
> (if I recall correctly).

That's a bit sloppy (but most of us do it from time to time).

> I thought I had picked up a subtle difference between "0 <= i <= k" and "i
> in [0, k]", it being a matter of usage: the former is used for contexts in
> which i ranges over the full interval [0, k], whereas the latter i is used
> specifically to imply a fixed value. Perhaps I am mistaken....

You are.

> Anyhow, would the following be synonymous with the above definition
> "For q in Q(n), let A(k, q) = {i in [0, k] : q | k+i}" ?

Yes, but it should be q | (k + i).

For this sort of thing we rely on the Axiom of Specification:

"To every set A and to every condition S(x) there corresponds a set B
whose elements are exactly those elements x of A for which S(x) holds."
-- Paul Halmos

So, to guarantee ourselves a new set we must /a priori/ have an
existing set and a "condition". We customarily shorthand Halmos' text
as:

   B = {x in A : S(x)}.

Signature

Paul Sperry
Columbia, SC (USA)

Reply to this Message
Frederick Williams - 07 Jul 2009 11:26 GMT
> > For q in Q(n), let A(k, q) = {i : 0 <= i <= k, q | k+i}
> > (if I recall correctly).
>
> That's a bit sloppy (but most of us do it from time to time).

Why?

> > "For q in Q(n), let A(k, q) = {i in [0, k] : q | k+i}" ?
>
> Yes, but it should be q | (k + i).

I think the brackets are redundant, after all (q|k) + 1 has no meaning.

Signature

Which of the seven heavens / Was responsible her smile /
Wouldn't be sure but attested / That, whoever it was, a god /
Worth kneeling-to for a while / Had tabernacled and rested.

Reply to this Message
Brian M. Scott - 07 Jul 2009 19:00 GMT
>>> For q in Q(n), let A(k, q) = {i : 0 <= i <= k, q | k+i}
>>> (if I recall correctly).

>> That's a bit sloppy (but most of us do it from time to time).

> Why?

Why is it a bit sloppy, or why do we do it?  I consider the
use of a comma instead of 'and', '&', or '/\' sloppy.

>>> "For q in Q(n), let A(k, q) = {i in [0, k] : q | k+i}" ?

>> Yes, but it should be q | (k + i).

> I think the brackets are redundant,

In general they are, and I'd probably omit them, but they
don't hurt and may improve readability.  I'd probably
suggest that beginners use them, to reduce the effort that
they have to put into interpreting notation and free up
attention for the actual content.

> after all (q|k) + 1 has no meaning.

It could by a previously established convention mean '2 if
q|k and 1 otherwise', but [[q|k]] + 1 would be *much* more
usual, and either would have to be explicitly set forth.

Brian
Reply to this Message
Frederick Williams - 07 Jul 2009 21:35 GMT
> >>> For q in Q(n), let A(k, q) = {i : 0 <= i <= k, q | k+i}
> >>> (if I recall correctly).
[quoted text clipped - 4 lines]
>
> Why is it a bit sloppy, or why do we do it?  

The first.

> I consider the
> use of a comma instead of 'and', '&', or '/\' sloppy.

Ok.

> >>> "For q in Q(n), let A(k, q) = {i in [0, k] : q | k+i}" ?
>
[quoted text clipped - 4 lines]
> In general they are, and I'd probably omit them, but they
> don't hurt and may improve readability.  

True.

Signature

Which of the seven heavens / Was responsible her smile /
Wouldn't be sure but attested / That, whoever it was, a god /
Worth kneeling-to for a while / Had tabernacled and rested.

Reply to this Message
Frederick Williams - 08 Jul 2009 15:44 GMT
> Why is it a bit sloppy, or why do we do it?  I consider the
> use of a comma instead of 'and', '&', or '/\' sloppy.

[...]

> > after all (q|k) + 1 has no meaning.
>
> It could by a previously established convention mean '2 if
> q|k and 1 otherwise',

If you can give (q|k) + 1 such a meaning then why not let , mean and?

Signature

Which of the seven heavens / Was responsible her smile /
Wouldn't be sure but attested / That, whoever it was, a god /
Worth kneeling-to for a while / Had tabernacled and rested.

Reply to this Message
Brian M. Scott - 09 Jul 2009 22:37 GMT
>> Why is it a bit sloppy, or why do we do it?  I consider the
>> use of a comma instead of 'and', '&', or '/\' sloppy.

> [...]

>>> after all (q|k) + 1 has no meaning.

>> It could by a previously established convention mean '2 if
>> q|k and 1 otherwise',

> If you can give (q|k) + 1 such a meaning then why not let
> , mean and?

Why, when there are perfectly good, well-known standard
alternatives?  Not that I'd recommend using parentheses as a
pseudo-Iverson notation, mind you, but I can imagine someone
reinventing this particular wheel: Iverson notation isn't
all that well known.

Brian
Reply to this Message
Axis - 07 Jul 2009 19:05 GMT
>> Paul,
>>
[quoted text clipped - 20 lines]
>
> That's a bit sloppy (but most of us do it from time to time).

In my usage, if I write the definition for which the context is i in [0,k],
I will persistently have to put in a '-1' when I use the argument (y-x)/2-1;
but with the definition the mathematicain wrote out, I can change the >= k
to >k, and it obviates this issue. So presumably the sloppy version is
better.....

Which brings me back to an old question. Recall that in a recent post I was
querying the use of calculations in
"C(x,y,q) = \/({A'((x+y)/2, r): r in Q(n) \setminus {q}} \/ F(x,
(x+y)/2)) /\ A((x+y)/2,q)) /\ [0, (x+y)/2-1]."

Well, I've got another set, which is the set, L(x,y,n), of all A(k,q) \cup
A'(k,q) such that q in P(n).
(Incidentally, A'(k, q) = {i : 0 <= i <= k, q | (k-i)}.
I think I ought to change the arguments in L(r, n). Better still, to use two
endpoints r and s that are formed from calculations using exclusively x and
y as variables.
For the definition I am keen to change r for L(r,s,n) to prod(Q(n))+(x+y)/2.
If I don't, my usage, substituting of k for prod(Q(n))+(x+y)/2, will become
unwieldy. What do you think I ought to do? When you said there is never any
excuse for using a calculation in place of a dummy variable in a definition,
what about unwieldiness in usage -- that's a good excuse, isn't it?

With thanks.
Reply to this Message
Axis - 07 Jul 2009 19:08 GMT
Correction -

"I can change the >= k
to >k, and it obviates this issue "

should read "...<=k to < k..."
Reply to this Message
Axis - 07 Jul 2009 19:36 GMT
> Well, I've got another set, which is the set, L(x,y,n), of all A(k,q) \cup
> A'(k,q) such that q in P(n).
> (Incidentally, A'(k, q) = {i : 0 <= i <= k, q | (k-i)}.
> I think I ought to change the arguments in L(r, n). Better still, to use
> two endpoints r and s that are formed from calculations using exclusively
> x and y as variables.

Indeed, this gives rise to an issue in itself. Faced with a choice between
using two arguments x and y in a term g(x,y,n), a value which makes use of x
and y exclusively by way of the expression (x+y)/2, and a term g(k,n) where,
in all usage of the term, (x+y)/2 is substituted for k, which would be
preferable?
I must say that, if it makes a difference, unwieldiness is a distinct issue
in what I am writing, what with a host of different arguments being used in
some of my sets (so much so that any instruction on the mathematical concept
of dependency - specifically on the matter of whether dependency needs to be
acknowledged, in the form of an argument, when it is implied by prior
definition - would be most welcome!).

WTIA for any assistance.
Reply to this Message
Paul Sperry - 08 Jul 2009 07:26 GMT
> >> Paul,
> >>
[quoted text clipped - 26 lines]
> to >k, and it obviates this issue. So presumably the sloppy version is
> better.....

Sloppy is _never_ better.

For you, [a, b] = {n in |N : a <= n <= b}.
Also, [a, b) = { n in |N : a <= n < b} is standard.
As is (a, b] = { n in |N : a < n <= b} and
(a, b) = { n in |N : a < n < b}.

If you have defined sets A(m, n) for integers m and n and if (X + y)/2
is an integer then it is perfectly OK to reference A((x + y)/2, n).

What is _not_ OK is to try to define \ab initio\ A((x + y)/2, n) in
terms of x and y and n.

I am utterly bored with these kinds of questions and am unlikely to
respond to any more. Read a Math book or two.

Signature

Paul Sperry
Columbia, SC (USA)

Reply to this Message
Axis - 05 Jul 2009 15:26 GMT
> One question about ordered pairs: is it admissible to have such a pair as
> (a,a)? And if one says "M,N \subset J", or maybe "subsets M and N of J",
[quoted text clipped - 7 lines]
>
> With thanks.

Any assistance on any of these matters would, as ever, be greatly
appreciated.
Reply to this Message
Paul Sperry - 06 Jul 2009 05:23 GMT
> > One question about ordered pairs: is it admissible to have such a pair as
> > (a,a)?

Of course. Have you never graphed a function?

> > And if one says "M,N \subset J", or maybe "subsets M and N of J",
> > is that prescribing that M and N are distinct subsets of J?

No

> > (Perhaps the
> > word I should use there is "unique", instead of "distinct"?).

Definitely not.

> > A final matter; Is it OK to define my Y(L) (or 'Y_L', as it was), as 'any
> > set of intervals of length L',

No - Y_1 could be {[1,2]} or it could be {[1,2], [2,3]} or it could
be ....

> > and then follow it with 'Choose L so that L
> >  > prod(K)?" It might seem to contradict the use of 'any'; but it's not
> > till a little later on that I introduce K.
[...]

Signature

Paul Sperry
Columbia, SC (USA)

Reply to this Message
Axis - 06 Jul 2009 13:57 GMT
>> > (Perhaps the
>> > word I should use there is "unique", instead of "distinct"?).
[quoted text clipped - 7 lines]
> No - Y_1 could be {[1,2]} or it could be {[1,2], [2,3]} or it could
> be ....

S if I say '*the* set of intervals of length L', presumably it is OK to
follow it with 'Choose L so that L < prod(K)...'?

I take it that the fact that you are not objecting to my use of L means it's
OK to use a length, and by implication, cardinality as an argument for a
function?
With thanks.
Reply to this Message
Axis - 06 Jul 2009 20:18 GMT
I asked a question 2 or 3 weeks ago, which went, if I define m = prod(J),
and then later preface a claim with 'For m in S...', can I be certain that
the reader will take it that I am redefining m, and not saying 'prod(J) in
S'?
The reply was, quite fairly, along the lines that I should do everything I
could to be clear. But unless there is some convention to settle this
matter, I can't see where the boundaries, that enable full
comprehensibility, lie.
For example, I have a set S(x,y,n) which is a set of sets, that I have
defined with the notation A(k, q) and A'(k,q).
If I say 'For A(k, q) in S(x,y,n)', can I be certain I will not be
interpreted as saying 'For r in S(x,y,n)'?
(Actually, perhaps that should be 'For R in S(x,y,n)', since R is a set.)

And as a related issue, if I write a definition that begins 'For q in Q,
let....', if in the next sentence, or the next-but-one sentence, I need to
use q in Q, do I need to introduce it a second time, with the repeat preface
'For q in Q'?

With thanks in advance.
Reply to this Message
Axis - 07 Jul 2009 18:50 GMT
>I asked a question 2 or 3 weeks ago, which went, if I define m = prod(J),
>and then later preface a claim with 'For m in S...', can I be certain that
[quoted text clipped - 16 lines]
>
> With thanks in advance.

Any further help on these things would be much appreciated....
Reply to this Message
 
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