User:Pyenos/January, 2007

Zakon's Basic Concepts

Basic Set Theoretic Concepts

Currently, I find it neccesary to learn the fundamentals of foundations of mathematics, in order to understand and correctly utilize mathematical symbols and notations. So, I'm teaching myself on the basics of set theory by reading books such as 'Basic math concepts, by Zakon' or ironically, the 'Mizar manual', which is not an introductory material to set theory, but rather an introduction to Mizar language used for formalizing mathematical proofs for the purpose of computer verification. I wish to admit, also, that I'm learning a few things on basics of set theory by reading relevant Wikipedia articles ;) --Pyenos 03:05, 26 January 2007 (UTC)

I have finished reading in 'Zakon series on mathematical analysis - Basic math concepts', sections 'Introduction. Sets and their elements' and 'Operations on sets'. --Pyenos 20:30, 28 January 2007 (UTC)

I have completed providing draft answers to 14 out of 20 questions of 'Set theory questions' in 'Zakon series on mathematical analysis - Basic math concepts'. I note that question 20 is about Russell's paradox, which I attempted to provide a contribution on Talk page of the corresponding Wikipedia article few days earlier, without the mastery of set theory concepts. It is an obvious logical step to attempt again to contribute to the Russell's paradox page after I have completed all of draft answers to 'Set theory questions', hopefully with a clearer solution. --Pyenos 04:36, 29 January 2007 (UTC)

Completed two more questions and now the progress is 16 out of 20 questions on 'Set theory questions' in 'Zakon series on mathematical analysis - Basic math concepts'. I found two things very interesting. First, if A AND B is denoted AB, A OR B is denoted A+B, and complement of A is denoted -A, then a logical statement like (A AND B)OR C can be worked at much like multiplication using distributive law, like this: $AB+C=(A+C)(B+C)$. Second, question 17 mentions symmetric difference of two sets, A and B, as $A\Delta B=(A-B)+(B-A)$. The wikipedia article mentions that it is equivalent to exclusive disjunction in binary operation (XOR). --Pyenos 16:09, 29 January 2007 (UTC)

I thought solving $A\Delta (B\Delta C)=A\Delta (C\Delta B)$ using assumed distributive property was amazing(which by chance of luck I had a guess at and succeeded for one case). But distributive property is not held for $A\Delta (B\Delta B)=A\Delta \emptyset =A$, $(A\Delta B)\Delta (A\Delta B)=\emptyset$ --Pyenos 16:10, 29 January 2007 (UTC)

I've learned that strictly distributive law applies only to two different operators. Anyways, I've finished 19 out of 20 questions on 'Questions in set theory' in 'Zakon's basic math concepts'. The next and last question, Q20 is on Russell's Paradox ;) --Pyenos 09:23, 30 January 2007 (UTC)

I've finished providing draft answers to 'Set theory questions' and now I'm reading 'Logical quantifiers' section. --Pyenos 10:12, 30 January 2007 (UTC)

I still have a confusion over how to represent the sentence "For all x in A, there exists x in B..." as a Venn diagram. --Pyenos 11:35, 30 January 2007 (UTC)

Limit of a sequence (x_1, x_2, ... , x_n, ... ) on $\mathbb {R} ^{+}$ is described by (using Zakon's notation)

$(\forall \epsilon >0)(\exists k)(\forall n>k)|x_{n}-p|<\epsilon$

It helps to translate this into longer form like this

$(\forall \epsilon |\forall \epsilon >0)(\exists k)(\forall n|n>k)|x_{n}-p|<\epsilon$

And to translate into a form which describes the processes by defining a function $\Phi [x]$ like this

$\Phi [x]:=(\forall \epsilon )(\epsilon \cap \epsilon >0\Rightarrow \Phi [x])$

$(\exists k)(k\cap \Phi [x])$

$(\forall n)(n\cap n>k\Rightarrow \Phi [x])$

$\Phi [x]:|x_{n}-p|<\epsilon .$

I'm not sure whether they are correct, but with this description I was able to represent the 'flow of information x' in my own way, using circles as domains of each part and shading of the circle to represent quantity of x and arrows or lines to represent flow of x from one domain to another, finally arriving at $|x_{n}-p|$.

With all this, I was able to draw positions of $x_{k},x_{n}$ with respect to $\epsilon ,\epsilon +p$ on $\mathbb {R} ^{+}$ line like this

0----------------- x_k -------- (-epsilon ---- -epsilon+p ---- x_n ---- epsilon ---- epsilon+p) ---> R^+

Which makes very clear what the original statement was describing.

In casual terms,

For each arbitrary neighborhood of positive reals, there is some point which is situated before it.

The negation in casual terms,

For each arbitrary point of positive reals, there is some neighborhood that doesn't exist which is situated after it.

It is as if there is a string with two points (x_k,x_n) in which some amount of string surrounding x_n is rolled in a circle and stuck back to the string without losing any length.

                  /---\ roll
                 *|   |* imagine (*) glued back to the main string without losing string length
                   \ /
*-------.---------/ . \--------->

This is certainly impossible in reals. --Pyenos 04:33, 31 January 2007 (UTC)

Domain and Codomain of xRy are

$D_{R}:=\{x|(\exists y)xRy\}$

$D'_{R}:=\{y|(\exists x)xRy\}$

I thought these are very interesting and it is plausible to represent flow of information x from domain to co-domain to domain and so on, as a diagram. I felt that even 'element of' is a kind of relation that links an object to a set. The concept of relation seemed to me to be closely tied to the concept of function. I imagined that D_R and D_R' are represented by two circles and let's say there is some x in D_R and some y in D_R'. The definitions above says that domain is a set of x such that y travels from D_R' to x in D_R via R. Conversely, codomain is a set of y such that it is linked by incoming x via R. So each elements of x and y are linked to each other bidirectionally. I imagined that R is a space that contains both D_R and D_R'. I'm guessing that R should be superset of both D_R and D_R'.

Inverse R^-1 is a space such that it contains y in D_R and x in D_R' and that each x and y are connected bidirectionally. --Pyenos 11:03, 31 January 2007 (UTC)