Course# 408 for 4th year 2009-10 Instructors: Shaheen Islam & Masum
Ahmed 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/matured – so 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
✔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
May 18/10 10:30 - 11:00 Instructor:
Shaheen Islam
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 (ℕ) and the set of square-numbers. Infinite sets Important things to point out (in future) notation and convention ℕ the set of natural numbers (প্রাকৃতিক সংখ্যা) W the set of whole numbers (পূর্ণ সংখ্যা) Masum
Ahmed: May 18, 2010May 19/10 10:45 - 11:30 Instructor: Shaheen IslamA ⊂ 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, 2010I 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................) May 29/2010, around 10 AM, Instructor: Shaheen IslamJune 5/2010, 11:30 AM, Instructor: Shaheen IslamI 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.
BT&F F
. 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 B and A, and the
output is a compound sentence B&A. And, having B and A as inputs into &, you get only B&A, not of course BA˅ : here lies the functionality of &. B
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 We will rewrite the truth-function calculation so that the two binary functions are shown to be distinct.
BT&ºF 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 ? 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 E .
)BE(A) &º(
E )BT&ºFF 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)
= B)E ()&º AE
()B^{n
}→ {T, F}
so that E (Ω( A_{1}, A_{2},
A_{3} ... A_{n} = Δ()E ( A_{1}), E ( A_{2}), E ( A_{3}), ... E (A_{n})) 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.domain: বলয় How about বস্তুগত নিঃসরন for material implication?] co-domain: সহ-বলয় range
আওতা truth-function সত্যাপেক্ষক truth-functional সত্যাপেক্ষকীয় |