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Mathematical Logic (course 408)

 

Course# 408 for 4th year 2009-10
Course Title: Mathematical Logic

Instructors: Shaheen Islam & Masum Ahmed
formal period: begins on 5th April 2010 ends on 5th Jan 2011

Actual period begins: after 20 April 2010

Course Content:

The aim of this course is to get the student familiar with some basic notions of mathematical logic: basic set operations, functions, relations... 

Requirements/Pre-requisites: Virtually nothing except your attentiveness

End result: What you will learn will make you skilled/maturedso that you will understand some basic notions of mathematics and computer science.


 

 


May 12, 2010, 10:30 - 11:15 Instructor : Shaheen Islam

We have discussed about three characteristics/rules pertaining to sets

1) there is a null or empty set; whereas with classes, categories (epitomized in Aristotle's Categorical Propositions) we don't think that there are empty categories or classes

2) x ≠ {x}

3) identity condition: (the axiom of extensionality)

  A = B  if both A and B share same elements


Important things to point out (in future)

 3) is not applicable to the notion of committees. The Academic committee and the Planning committee may have same members, but yet the two committees are distinct

May 15, 2010 Instructor: Masum Ahmed


It was very elementary class about Mathematical Logic. I noticed students are very interested about the course..


May 18/10 10:30 - 11:00 Instructor: Shaheen Islam

[A class-routine is officially given on this day]

difference between the notion of committees and that of sets.
memebership, subset, with notations; the notion of subset in terms of material implication,
set-identity in terms of subset relation and material equivalence.
proper subsets
complimentary/relevant  story:
Galilieo's puzzle regarding the one-one correspondence between the set of natural numbers () and the set of square-numbers. Infinite sets

Important things to point out (in future)


notation and convention
the set of natural numbers (প্রাকৃতিক সংখ্যা)
W the set of whole numbers (পূর্ণ  সংখ্যা)

Masum Ahmed: May 18, 2010


May 19/10 10:45 - 11:30 Instructor: Shaheen Islam


A ⊂ B     A is called "subset"(উপসেট) and B is called "superset" (অধিসেট)

There is only one empty set in standard set theory. This follows from  the axiom of extensionality


relation and property,   arity 


Masum Ahmed: May 19, 2010

I got the official class routine today.
 I have two classes: Wednesday and Thursday
Accordingly, I went the class room today and discussed on SET AND SUB -SET, the introductory topic of the syllabus.
(Definition of Set, Notation................)
 
function: functionality




May 29/2010, around 10 AM, Instructor: Shaheen Islam


June 5/2010,  11:30 AM, Instructor: Shaheen Islam



I wanted to explain the following content in the above two classes (29th May and 5th June)

key concepts: function (অপেক্ষক), domain (বলয়), co-domain (সহ-বলয়), range (আওতা), truth-function (সত্যাপেক্ষক), truth-functional (সত্যাপেক্ষকীয়)

A function is a sort of machine/process that gives you an output out of some inputs. The output has to be determinate or consistent – in the sense that there cannot be different outputs (say at different times) out of same inputs. Plus (+), for example, is a function. We put 3 and 2 as inputs (into this function), and we get 5: we write 3+2=5. But if we had 5 sometimes, and 4 other times, then plus would not be counted as a function; we would say then that the function (plus) has lost its functionality.

You must be familiar (in your earlier courses) with this truth-value calculation:

A&B
T&F
F


A lot of things have happened here; you might have not noticed them – as you did the calculation mechanically. You can see & there as a function. But, in fact there are two different functions which were denoted by the same sign “&”. One function is called sentential connective or just connective, as it connect or join two sentences A and B. The other function is called truth-function, for it takes two truth-values T and F as inputs and having those truth-values it gives you F as an output. Yes, this – what we call to be a truth-function – is a function; the truth-table shows you that – there are only a unique result/output for each pair of truth-values : TT, TF, FT and FF give you T, T, T and F respectively. &, the sentential connective is a function too; the inputs here are sentences A and B, and the output is a compound sentence A&B. And, having A and B as inputs into &, you get only A&B, not of course A˅B : here lies the functionality of &.


Let us disambiguate. We will keep on “&” to designate the connective, but for the truth-function we will use &º. & and &º are, of course, different; for the connective & gives you a compound sentence (or proposition) out of some input sentences, whereas the truth-function “&º” gives a truth-value out of some input truth-values. They are, of course, similar in some respects. For example both the functions need two inputs to produce an output, they are then called to be binary functions, in other words – we say – their arity is 2. But, right now we are interested in their difference not in their similarity. How do they differ? Answer: They differ with respect to the types of inputs and outputs. The connective works in an environment of propositions, whereas the truth-function works in an environment of truth-values. This much of information, however, is not enough. What else do we need? We will go back to the truth-value calculation.


We will rewrite the truth-function calculation so that the two binary functions are shown to be distinct.

A&B
TF
F

Our explication is not complete yet. There lies another function; so far it was hidden. This function gives you a truth-value – as the output – from any input sentence; the sentence might be any sentence – be it simple or complex. We will call this function extension function, which extract out just the truth-value of a sentence. Let us denote it by E . For example,


E (3=1) = F ,             E (3=3) = T ,           E (India had been colonized for around 190 years) = T


Now, how can we describe the environment pertaining to E ? It is neither an environment of propositions nor an environment of truth-values. Rather it is that of a mixture of both propositions and truth-values. A precise description of the environment seems to be this: E takes a proposition as input and gives us a truth-value as output. A compact way of describing this is as follows:

propositions truth-values


In diagrams:

                                                              

The class or set of propositions, which can actually be inputs – in the sense if you feed the input in E then you must get a unique output – of E is called the domain (বলয়) of E . And the set of truth-values – {T, F} – which are potential – in the sense, may become but not necessarily – outputs of E is called the co-domain (সহ-বলয়) of E;  and the set of truth-values which will be actually an output – having an appropriate input – of E is called the range of E .

Let us comeback to our truth-value calculation. Now we will show the extension function E .


E ( A & B )
E(A ) E( B )
T
F

F

The calculation is possible because implicitly there has been the following equation, (which enables us to go to the second line from the first one) 


E (A&B) = E (A)&º E (B)


And, this very equation makes the sentential connective &  to be truth-functional. In general a sentential connective, say Ω which has arity n ( practically n cannot be even greater than 2), is said to be truth-functional if there is an n-ary truth-function  Δ: {T, F}n {T, F} so that

E (Ω( A1, A2, A3 ... An ) =  Δ(E ( A1), E ( A2), E ( A3), ... E (An))

বাংলা পরিভাষা:

set:                                         সেট

identity:                             অভিন্নতা

unit set or singleton :     একক সেট


implication :                         নিঃসরন    ?


material implication:         বস্তুগত নিঃসরন  ?
[We are not very happy with these  terminologies though they sound better than the traditional ones.
How about বস্তুগত নিঃসরন for material implication?]


domain:                                 বলয়     
co-domain:                           সহ-বলয়   
range                                     আওতা
truth-function                      সত্যাপেক্ষক
truth-functional                   সত্যাপেক্ষকীয়


Subpages (1): Abstract Truth Functions
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