Download Topics in Mathematical Science

Topics in Mathematics
Credits: 4 Credits
Course Coordinator: Prof. Manohar Lal
OBJECTIVES
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Computer Science is, essentially, constructive mathematics with utilitarian (efficiency/cost /feasibility /reliability etc)
considerations of the proposed solutions. The enterprise of computing, i.e., design and development of software, revolves
round the concept of algorithm. Program, i.e., an algorithm expressed in a way so as to be understood and executed by a
computer system, is a mathematical construct. The construct may be a program in a programming language ( a
mathematical system itself) like C, LISP or PROLOG or it may be a Lambda-expression, a partial recursive function or
some formal expression in any of the newly emerging mathematical computational models (of various biological and other
systems) like neural network, evolutionary computation, membrane calculi and quantum computation etc..
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Thus computer program in any of the formalisms, being itself a mathematical entity, requires both mathematical knowledge
and mathematical (including logical) skills for its construction/ development. Further, in view of the utilitarian aspects of
computer science, the constructs/ programs, as proposed solutions, need to analysed from the
feasibility/efficiency/cost/reliability aspects. The analysis, again requires mathematical tools and techniques.
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The above-mentioned required mathematical knowledge is quite diverse and deep, and demands sufficient knowledge of
basic mathematics and mathematical reasoning skills. The purpose of the course is to teach this basic mathematics and hone
these mathematical reasoning skills.
PROPOSED SYLLABUS
I.
Ideas & and their contributions (20 %):
o Famous problems/ conjectures (solved/unsolved):Three classical problems, fifth postulate,
Fermat’s last theorem, four-colour problem, Goldbach’s conjecture and other conjectures in
mathematics Paradoxes: Zeno’s, Russell’s, Liar’s, Barber’s, Berry’s, Grelling-Nelson’s,
Richard’s, Bertrand’s etc. and other paradoxes from set theory, logic, statistics and probability;
argument, Fallacious arguments; Mathematical and logical Puzzles; Concept of Proof.
o
Contemporary Schools of thoughts (philosophy of mathematics): Realism, Logicism,
Formalism and Intuitionism (constructivism & finitism)
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Concept of Proof & Proof techniques: By contradiction, by counter-example, by cases,
contrapositive, proof of without word. Diagonalization Principle, pigeon-hole principle,
principle of mathematical induction (PMI), stronger PMI, principle of structural induction.
o
Controversies and New Trends: Pythagoreans and irrational numbers, Cantor’s Theory,
Berkley’s objection to foundations (use of infinitesimals) of calculus, Fifth Postulate, Cantor’s
theory of infinite sets. Proof by computers, Experimental mathematics , Chaitin and others for
accepting statements as mathematically true which have not been found false (e.g. Goldbach
conjecture).
2. Statistics and probability theory (30%):
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Axioms of probability, Baye’s formula, expectations of random variables, jointly distributed
random variables, conditional expectation, some applications – a list model, a random graph.
Limit theorems, random number generation, simulating continuous random variables,
Probability Distributions: Binomial, Poisson, Normal
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3. Counting principles & mathematical structures (20%)
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Generating functions, recurrence relations, linear recurrence relations with constant coefficients,
homogenous solutions, particular solutions, total solutions, solution by the method of generating
functions.
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Introduction to algebraic structures groups, fields, lattices and boolean algebra.
4. Mathematical Logic: (30%) Propositional Calculus
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Propositions, well-formed formulas, Semantics/Meaning in Propositional Logic, tautologies,
equivalence of formulas, duality law, normal forms, inference theory for propositional calculus;
Applications of propositional calculus
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Predicate calculus: syntax in predicate calculus: predicates, functions, free and bound variables,
prenex normal form (PNF), (skolem) standard form, inference theory of predicate calculus.
Applications of predicate calculus
REFERENCE BOOKS:
1.
Discrete Mathematics & Its Applications (Fourth Edition) by Kenneth H. Rosen ( Tata
McGraw-Hill Publishing Company, 2003)
2. Introduction to Probability Models, 8th ed., by S.M. Ross (Academic Press, 2004)
3. Applied Statistics and Probability for Engineers (Fourth Edition) by D.C. Montgomery &
G.C. Runger (Wiley-India, 2007)
4. Introduction to Mathematical logic (Fifth Edition) by Elliot Mendelson (CRC Press, 2010)
5. Symbolic logic: Classical and advanced systems, II Edition, by Harry J. Gensler (PreticeHall 2010)
6. Computability: Computable functions, Logic, and the Foundations of Mathematics (Second
Edition) by R.L. Epstein & W.A. Carnelli (Wadsworth, 2000)
7. Experimental Mathematics in Action by Bailey et all (A.K.Peters, 2008)
8. The Lady or the Tiger and Other Logical Puzzles by Raymond Smullyan (Penguin Books,
1982)
9. Concurrent Zero-Knowledge by Alon Rosen (Springer, 2006)
10. Statistics and Truth: Putting chance to work by C.R.Rao (F.R.S), Ramanujan Memorial
Lectures (CSIR, 1989)
11. Exploring Randomness by Gregory J. Chaitin (Springer, 2001)
12. Discrete Thoughts: Essays on Mathematics, Science, and Philosophy by Mark Kac, GianCarlo Rota, and Jacob T. Schwartz, (Brikhauser, 1992)
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REFERENCE COURSE MATERIAL
1.
IGNOU’s course material for MCSE-003 and for MCS-033.
JOURNAL ARTICLES
WEBSITE REFERENCES
RELEVANT VIDEOS
ASSIGNMENTS
HANDOUTS / SLIDES USED FOR THE PRESENTATIONS
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NEWS / ANNOUNCEMENTS
EMAIL: [email protected]
131, School of Computer and Information Sciences (SOCIS),
C-Block, First Floor, IGNOU Academic Complex,IGNOU,
Maidangarhi, New Delhi – 110068. Tel: +91-11-29572906
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