Defining Set Configurations with Properties and Functions

[zrek]

Please help me to define correctly, in the language of mathematics, the configuration of sets shown on the picture.

Homework Statement

I'd like to define the following rules:
U is a set with infinite members.
L is a list or set of properties. Every property (Ls1, Ls2 ... ) have a value ( 0,1,2,...P )
Every member of U is defined by its unique property list. Every member is paired with one and only one value from each of the Ls1,Ls2... properties.
The U contains i+1 subsets.
F is a list of functions, or it is the set of subsets of U.
Every subset of U is paired with a function in F, except one, which contains all of the members which are not in any subsets defined by F.

The subsets of U are determined the following way:
An Xk member of U is in subset Fj, ( 1<=j<=i ) if the function Fj outputs "true" after processing the Ls1, Ls2 ... values of Xk.

For example:
Properties of X0 are:
Ls1: 3
Ls2: 2
Ls3: 5
...
Since only the function F1 returns "true" for this input, the X0 is solely in the F1 subset of U.

The Attempt at a Solution

I have difficulties even to correctly define the L set and the members of U.

Would you please help me to start somehow?
Thank you in advance.

 

[mfb]

What do you mean with "correctly define the L set"?

You can let each element in U be a function u_j: M -> {1,2,...,P}
Where M is the set of properties.

 

[haruspex]

zrek, I find your notation very confusing. Do you mean that the Ls are functions from U to {1 ... P}, and that given x in U the vector (Ls1(x), Ls2(x)...) is unique?

 

[zrek]

What do you mean with "correctly define the L set"?

I mean I don't know how to describe it in the language of mathemathics, with clear formulas.
Is L a set at all? Or maybe a list? (The index of the elements of the L is an important data, I think)
L is only used to define the requirements of the elements of U.
The requirement of the elements of U is that to have infinite values, one from each member of the L.
For example u₀ is a zero element, if all of its values are 0. u₀={0,0,0, ...} Is this a good notation for it?
Let's say u_mx is a "max element", if if all of its values are P. u_mx={P,P,P, ...}

You can let each element in U be a function u_j: M -> {1,2,...,P}
Where M is the set of properties.

I don't want to say that the elements of U are functions. The functions describe the subsets of U.
There are elements in U which are can not be accepted by any of the functions in F.

Thank you for your help, mfb.

 

[zrek]

zrek, I find your notation very confusing. Do you mean that the Ls are functions from U to {1 ... P}, and that given x in U the vector (Ls1(x), Ls2(x)...) is unique?

Sorry, you are right, it is confusing, I'm not experienced describing sets, I'll try to learn it.

I'm uncertain of the meaning of "vector". I know that v(x,y) is a vector, if x and y are real numbers. In our case the index values are discrete. Can we say that u(a,b) is a vector, if "a" and "b" can have only discrete values from [0,1,2...P] ? If this is ok, then it makes easier, we can say that U is a set of vectors.

I found a good analogy. Please consider the followings as an example:

U is a set of infinite many marbles.
The marbles have properties. All of the property types are listed in the list L. (there are infinite many of them)
Every property value is "normalized" between 0 and P.
The first property type (Ls1) is the weight. The weight of a marble can be between 0 and P.
The second property type (Ls2) is the roundness. The roundness also a value between 0 and P.
The marbels are colorful. There are 3 color component: R,G,B.
The third property type (Ls3) is the R component ...
(...and so on, there are infinite many property types.)
Every marble have to have a value for every property type.

Now we can create subsets, there are i+1 subsets.
For example we can create a function that determines whether or not a marble is heavy.
F1 function outputs true for those marbles with the Ls1 property is more than P/2. F1 function describes the "heavy marbles" subset of marbles.
F2 function outputs true if the Ls3 value of a marble is P, and the Ls4 and Ls5 (G,B components) are 0. The F2 determines the "pure red" subset of marbles.
(...)
There are finite number of functions.

There are marbles which will be accepted by one function.
There are marbles which will be accepted by several functions.
There are marbles which will be not accepted by any of the functions.

I hope that this analogy makes it clear.

Thank you for your help!

 

[mfb]

I don't want to say that the elements of U are functions.

They do what functions are doing. The description will become easier if you call them functions.

The functions describe the subsets of U.

Those are different (types of) functions.

Can we say that u(a,b) is a vector, if "a" and "b" can have only discrete values from [0,1,2...P] ?

With mathematical vectors this is a bit tricky (it can be tricky to find a corresponding vector space), with computer science vectors it is no problem.

 

[haruspex]

I'm uncertain of the meaning of "vector". I know that v(x,y) is a vector, if x and y are real numbers.

Not quite. It's the (x, y) that's the vector. v(x,y) looks like a function of two variables (or, equivalently, a function of the vector (x, y)).

With mathematical vectors this is a bit tricky (it can be tricky to find a corresponding vector space)

The correct term is a 'tuple' (http://en.wikipedia.org/wiki/Tuple). Some allow tuple as a species of vector; others, as you say, require vectors to live in a vector space.

L is a list or set of properties. Every property (Ls1, Ls2 ... ) have a value ( 0,1,2,...P )

The notation for that is $Ls_j:U\rightarrow \{0,...,P\}$. (In words, "The function Ls_j is a map from U to the set {0,..,P}")

we can say that U is a set of vectors.

No, U is not a set of vectors. U is your plain set of things x. Given an x in U, you can construct the (infinite) tuple L(x) = (Ls1(x), Ls2(x), ... ). Given distinct x, y in U, $L(x) \neq L(y)$, right?
I'll call the set of all such tuples T. T = {(t₁, t₂, t₃, ...}:t_i ∈{0, ... , P}}. So $L:U\rightarrow T$.

The U contains i+1 subsets.

U is infinite, so has infinitely many subsets. I assume you mean there is some particular collection of subsets A of interest. |A|=i+1.

Every subset of U is paired with a function in F, except one, which contains all of the members which are not in any subsets defined by F.

Every member of A is paired... etc.
You have a choice of how to define F_j. For clarity I'll write F' here for one of the choices.
You can have them as functions of x in U
$F'_j:U \rightarrow \{ true, false \}$
or as functions of the tuples.
$F_j:T\rightarrow\{true, false\}$
These are connected by $F'_j(x) = F_j(L(x)) \forall x \in U$. I.e. F'_j is the composition of F_j on L.
For each F'_j, you can define a subset A_j of U by $A_j = \{x \in U: F_j(x) = true\}$. (Or, more succinctly, $A_j = \{x \in U: F_j(x)\}$.)
This is sometimes written as an inverse function: $A_j = {F'}_j^{-1}(true)$, but it's a slight abuse of notation (http://people.clas.ufl.edu/groisser/files/inverse_images.pdf).
Note that the A_j are not necessarily distinct.
Then $A = \{A_j\} \cup \{U- \cup A_j \}$.

Hope this helps.

 

[zrek]

Mfb, haruspex, thank you for the answers. I got so many information, I'd like to thinking about it (few days) before my next question.

 

[zrek]

Thank you for your help Mfb and haruspex, finally I think I figured out what to do.