CS2100: Discrete Mathematics Introduction John Augustine - - PDF document

CS2100: Discrete Mathematics

Introduction John Augustine augustine@cse Office: BSB 314 www.cse.iitm.ac.in/∼augustine/cs2100 odd2012 Teaching Assistants Sangeetha Jose William Moses

• S. Sumathi

CS2100 (Odd 2012): Introduction to CS 2100

What is Discrete Mathematics?

It is NOT continuous mathematics. I like to think of it as the study of puzzles. Discrete mathematics deals with discrete structures that lack continuity.

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Some examples will help us differentiate between continuous and discrete mathematics. How would you classify the following?

• A ball falling down moves smoothly or continuously

down.

• Two balls are to be placed in two distinct bins

among a total of n bins. How many ways can this be achieved?

• The science of planetary motion.
• The study of pollutants in water.
• You have a list of people and some idea about who

are friends and who are not in talking terms. You want to seat them in a banquet so that everybody is happy.

• How does the frequency vary when you pluck a

guitar string and slowly tighten the tuning knob?

• In a piano, you play a chord by simultaneously

playing two or more different notes. How many three-note chords are possible in an 10-key piano?

• How does the computer store the number π?

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Finding your way through a maze

A B

How many paths are there from A to B? In general, given a road network, we can ask:

• Is there a path from some point A to another point

B?

• If it exists, report a path.
• What is the shortest path?
• How many paths are there?
• What is the longest path?

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Course Syllabus

The material we cover will include: Mathematical Reasoning. Prepositions; negation, disjunction and conjunction; implication and equivalence; truth tables; predicates; quantifiers; natural deduction; rules of inference; methods of proofs; use in program proving; resolution principles; application to PROLOG. Set Theory. Paradoxes in set theory; inductive definition of sets and proof by induction; Peono postulates; relations, representation of relations by graphs; properties of relations; equivalence relations and partitions; partial orderings; posets, linear and well-ordered sets. Graph Theory. Elements

• f

graph theory, Euler graph, Hamiltonian path, trees, tree traversals, spanning trees.

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• Functions. Mappings;

injections and surjections; composition

• f

functions, special functions, pigeonhole principle, recursive function theory. Abstract Algebra. Definition and elementary properties

• f groups, semigroups, monoids, rings, fields, vector

spaces and lattices. Some more set theory. Finite and infinite sets; cardinality, countable and uncountable sets. Furthermore, I hope to cover some additional topics in combinatorics and probability theory.

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Connections to Other Computer Science Courses

Digital Logic Design. Theory of Computation. Data Structures and Algorithms. Operating Systems. Database Systems. Computer Networks. . . .

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