\(\def\finv#1{ f^{-1}(#1) }\)

0 Prereq and Prelim

0.1 Logic

• \(A \implies B\)

0.2 Set and Classes

• DEF equals, in
• Statements/Properties/(Things which are true) equals

• \(A=A\)
• \(A=B \implies B=A\)
• \(A=B \land B=C \implies A=C\)

• Properties between equals, in
  • \(A=B \land x \in A \implies x \in B \)
• Equivalent condition membership,=

• \( (x \in A \iff x \in B) \implies A=B \)
• What about other direction? Property btw =,membership with reflexivity

• DEF Set: class A is a set iff \(A \in B\) for class B
• REM Set Paradox
• \(M = \{ X | X \not\in X \}\)
• \(M \in M\) or \(M \not\in M\)
• How does the distinction of proper class and set avoid the paradox?
• We are essentially saying, M does not belong in this discussion because it cannot be a member of a set
  • M cannot be X
• What does that mean for \(M \in M\) or \(M \not\in M\)?
  • It says both these are false and, more importantly as a result of \(M \not\in M\), M cannot be in M?

• This seems alot more clearer than it did when I first read it
• Koldobsky, to learn it and forget it
• To approach the problem with a fresh mind, you have no idea how important this is.
• How to get better at forgetting. Don't take yourself too seriously

• Notice that this paradox is playing off of the \in to draw a contradiction

• The paradox is of the form, \(A \iff \neg A\)

• \( A \implies \neg A\) This statement is false. Because when hyp is true, the conclusion is false

• DEF Subclass

• Another relation between classes, built around implication and membership
• REL equality and issubclass
• Operation: union, intersection, relative complement
  • DEF A union B := \(\{x | x \in A or x \in B\}\)
  • DEF A inter B := \(\{x | x \in A and x \in B\}\)
  • DEF A minus B := \(\{x | x \in A and not x \in B\}\)
  • DEF not A := G-A
• Instance: Null set, family of set, set of all subsets

• Adjective: Disjoint

• not(AUB) = G-(AUB) = {x | x in G and not in (AUB)}

• ?? x not in (AUB) = x in G-(AUB)

• Ok, we just introduced alot of new names and things

• What statements describe their relations and properties?
• Let us first look at what each operation/thing exactly is

• How would we go about listing all possible combinations?

• x@n

• x@!x
• !(x@y)
• (x@y)#z

Sets

• Null := S
• U,I,- := SxS -> S
• P := S -> S
• = := SxS -> st

Not for sets?

Binary

• AND,OR,->,= := bxb -> b
• NOT := b -> b

Statements are algebras of binary variables

Statements

• AND,OR,->,= := stxst -> st

• NOT := st -> st

• Examine the relationship between these and set operation.

• Construction of Not relationship for sets

How to organize findings at high level and the work that went in?

• Organize the findings at high level,
• Organize the work patterns at a high level
• And forget it, dont hold on to too much

Organization of ideas, heuristic

• HEUR What am I looking at, at a very abstract level?
• HEUR Is there a clear way to define all the combinations of things? How big should the combinations be?
• HEUR Negation Consider the negation. a and !a
• HEUR Relations of a,b: a=b, a~b
• NOTE HEALTH&MENTAL I am pretty sure my need to reboot my brain is due to chemical imbalance/reinforced by bad reward loop. Which is why I think the advice of not taking oneself seriously is super important.
  • What to do when you find yourself in a bad state?
  • Create new state. Formed by the same mechanism that you arrived to the bad state.
  • Identify bad feedback loops and stop em in the future.
    • Result based reward vs Effort/instrinsic value based reward
• HEUR Examples
• HEUR Questions
  • What do questions and examples do?
    • Examples are a bit clear. It is an estimation or a phenomenum. It is a specific thing.
      • Focuses attention on specific things.
    • Questions It asks for an answer. directs attention. Can be more open ended.
      • Advance the current attention towards another goal.
• HEUR Consider alternate forms of the same object
  • Question-noun, question,ask-verb, questionable-adj
• HEUR Negation Example, Negation Question
• HEUR What is the most abstract whay to look at

• HEUR What do you want? Specific examples. A model which satistfies the different examples.

• What I as primarly dealing with was imagination/intuition/examining the properties and consequences

  • Examine the pros/cons

• HEUR Goal oriented thinking.

  • Gives you adrenaline/drive proximity to your goal
  • Cannot overdo it. Needs rest.

• DEF 2 statements are equal. f1(x,y,z) f2(a,b,c)

  • f1=f2 := f1(x,y,z,a,b,c) = f2(x,y,z,a,b,c)
    • In this approach you added more variables
  • f1=f2 := f1(x,y,z) <-> f2(a,b,c)
    • if f1 true then f2 true
    • if f2 true then f1 true
    • Since !(f1 true) is (f1 false), and using the contrapositives, A<->B !A<->!B
    • Does this definition give us the desired properties of equality?
    • A=A, A=B->B=A, A=B&B=C->A=C

• Transitive(implication) (A->B,B->C) -> (A->C)

• Associative(equality) &,| (A&B)&C=A&(B&C)
• A&B->A,B
  • A&B->A, A&B->B
• A = T&A,T|A

• Knock out

  • T&B = B
  • F&B = F
  • T|B = T
  • F|B = B
  • ??? Woah, why is &,| similar A B A&B A|B T T T T T F F T F T F T F F F F
  • Its beautiful, the way it just works...

• Substitution A=B->A@C=B@C

• F(A) and !A->!F(A). Then A

  • T=A|!A

• HEUR Statement/Argument simple and general as possible

• DEF Implication Linking. A, as a consequence we get B

  • A->B
  • ??? How do we formally formulate this?
    • Suppose A, and you find B to be true. Then you can write A->B
    • What do you mean by find?
      • (Go back to the LOGIC TABLE representation)
        • Is there a simpler representation instead of a full table?
        • (describes some general property of how the table is formed)
          • Table has rows, we are interested in values of 1 row, given certain values
          • values on each cell can be T or F
          • writing A->B could be describing the behavior of the whole table...
            • namely that when A is T, B is T over all the rows in the table
      • (Another functional/mapping representation)
        • Do we really need this 'row' can we just work with the variables?
        • Well, fundamentally the different variables form a table...?
        • well defined: each domain has single output -> the inverse is well defined
          • is there a better word? unique
          • In your table analogy, the rows have this property
            • Good, we isolated this critical property

• SHOW A&(B|C) = X

  • ->
    • APPROACH S1 -> A,B -> S2
    • A&(B|C) -> A,(B|C)
    • A=T, B=T&B -> B=A&B
    • B=A&B, C=A&C -> (B|C) = (A&B)|(A&C) = X
  • <- (NOTE My guess is that this direction is bit more complex)
    • (A&B)|(A&C)
    • Suppose !A. Then our statement becomes (F&B)|(F&C) = F|F =F, which cannot be.
      • ??? Why cannot be and consequently A=T? (Explain this to me again, this is quite fundamental)
      • You must make a distinction on what stating A means, ??? Why is it done in this style? what are the consquences of doing it in this style?
      • F(A). Suppose !A -> !F(A) ??? so the question is what is the gap that takes us
      • contrapositive gives F(A)->A. (Dang really? this -> is really powerful) (Descriptor of LOGIC TABLE?)
      • NOTE: there is an alternate argument interpreting F as a mapping. well-defined properties as well
        • ??? Does it mean these two ideas are related?
        • Why ofcourse. The well-defined is what we need to form A->B=!B->!A
    • This means A=T. Then our statement becomes (T&B)|(T&C) = B|C
    • So we have A,B|C = A&(B|C)

• If you dont follow through what you believe in, and give up when its hard, what trials can you overcome?

• You will fail many times in life, when working on a difficult thing.

  • If you stay down and don't get up, you will never improve.

• Don't loose sight of the forest for the trees

• Don't take yourself too seriously

  • Do not compare yourself to others. Focus on What YOU need to do to get to the next level

• What pleasure do you take? out of thinking about thses things?

  • Wouldn't you like to understand the motivating thoughts behind Groendick?
  • Or what he meant by a house and tools and framework?
  • I can only really experience these things when I am able-minded huh...
    • It's more of an experience than a achievement...

• HEUR It might be worthwile to write some key motivating examples to guide the thought process

• OH, don't i have... the lin alg stuff or alg stuff on a paper somewhere...
• HEUR Organize/Summarize your findings
  • I must admit, this is a key area which I have neglected over the years

Functions

• class A,B domain and range
• assignment from domain to range

• f:A->B a->f(a)

• These representations are binary in nature

• AA, a->a

• does x->y represent an object?

• f(a) does, but this is not binary
• f(f(a))
• ??? Consider the notational difference of a->b and f(a)
• x->y->z
• g(f(a))
• operation on functions

• commutativity in a restricted system

• x->y, x<-y

• Inverse \(finv{S}\)

• \(A \subset B\)

• HEUR I feel overwhelmed when so many new ideas come and I cannot think of them carefully

  • It's a familiar feeling, when there is too much to handle
  • I just end up rushing through the important stuff
  • Often times I have given up... or just had forgetten or repressed. Why does this happen?
  • How can I over come?
  • Its ok to be wrong or to miss things.
    • Try again and again until satistfactory
    • Try in an earnest manner, your heart must be in it.
    • Try whole-heartedly, towards a good direction
  • Be comfortable with an incomplete understanding.
    • As a student, you'll run into many abstract definitions: topological spaces, metric spaces, manifolds, the various structures of abstract algebra, and so on.
    • In math, a definition is what it is, no more and no less. While it's very good to develop an intuition about these various objects, there's a danger of looking for some deeper meaning that just isnt there.
    • John von Neumann once said, In mathematics you don't understand things. You just get used to them.
    • I don't advise taking this quote too literally, but I think it reinforces the point of not worrying about understanding everything right away.

• Time pressure

• Fall in love with the process

• Be mindful of your motivation

• Two Phase:

  • On-Hand Chaotic Work Phase
  • Relaxed Quiet Review Reminiscence Phase

• Produce the best work in a chaotic environment?

• Aural, visual, kinesis, read, writing, thinking
• visual, kinesis, aural