University of Aberdeen, Computing Science CS3511 Discrete Methods Kees van Deemter

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University of Aberdeen, Computing
ScienceCS3511Discrete MethodsKees van Deemter

• Slides adapted from Michael P. Franks Course
  Based on the TextDiscrete Mathematics Its
  Applications (5th Edition)by Kenneth H. Rosen

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Practicalities

• Lectures
• Mon 2PM (MT6)
• Tue 11AM (Old Senate Room)
• Practicals
• Tue 1-3PM (MT011)
• Tue 3-5PM (MT011)
• Dont go by this info alone, since things may
  change. Consult web page at least twice per week,
  and watch email to level3-students

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Key resources

• Lectures based on Michael Franks slides(Univ.
  of Florida)
• Were not using all his lectures
• Many changes in those that we do use
• Lectures will appear on the web
• Book Ken Rosen, Discrete Mathematics and its
  Applications. 5th or 6th edition. Then e.g. (5th
  ed.)
• http//www.mhhe.com/math/advmath/rosen/r5

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Other practical issues

• AssessmentStandard arrangments
• 75 exam
• 25 cont. assessment
• Arguably, this course comes too late in your
  curriculum
• You will be somewhat familiar with much in this
  course
• A little knowledge is a dangerous thing

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Module 0Course Overview

• A few general slides about the subject matter of
  this course.
• 14 slides, ½ lecture

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What is Mathematics, really?

• Its not just about numbers!
• Mathematics is much more than that
• These concepts can be about numbers, symbols,
  ideas, images, sounds, anything!

Mathematics is, most generally, the study of any
and all certain truths (about any kind of
concepts)
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Uses of Discrete Math

• Starting from simple structures of logic and set
  theory, theories are constructed that capture
  aspects of reality
• Physics
• Biology (DNA)
• Common-sense reasoning (logic)
• Natural Language (trees, sets, functions, ..)
• Anything that we want to describe precisely

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So, whats this class about?

• What are discrete structures anyway?
• Discrete (? discreet!) - Composed of
  distinct, separable parts. (Opposite of
  continuous.) discretecontinuous
  digitalanalog
• Structures - Objects built up from simpler
  objects.
• Discrete Mathematics - The mathematical study
  of discrete objects and structures.

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Discrete Mathematics

• When using numbers, were more likely to use N
  (natural numbers) and Z (whole numbers) than Q
  (fractions) and R (real numbers).
• Reason Q and R are densely ordered
• This notion can be defined precisely.
• Some DM notation will help (Notation should
  become clear later on.)

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• ?Q,lt ? is densely ordered because?x?Q ?y?Q
  (xlty ? ?z (xltz zlty) )
• (if xlty then there exists at least one z
  in between)
• Opposite of densely ordereddiscretely ordered

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• Yet, Q and R can be defined in terms of discrete
  concepts (as we have seen)
• This means that Discrete Mathematics has no exact
  borders
• Different books and courses treat different topics

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Discrete Structures Well Study

• Propositions
• Predicates
• Proofs
• Sets
• Functions
• (Orders of Growth)
• (Algorithms)
• Integers
• (Summations)

• (Sequences)
• Strings
• Permutations
• Combinations
• Relations
• Graphs
• Trees
• (Logic Circuits)
• (Automata)

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Some relevant notations

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Discrete Math for computing

• The basis of all of computing is manipulations
  of discrete structures represented in memory.
• DM is the conceptual foundation for all of
  computer science.

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Some examples

• Algorithms data structures
• Compilers interpreters.
• Formal specification verification

• Databases
• Cryptography
• Digital circuits
• etc.
• DM is relevant for all aspects of computing!

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Plan of Course (as per Rosen 5th ed.)

• Logic (1.1-4)
• Proof methods (1.5)
• Set theory (1.6-7)
• Functions (1.8)
• (Algorithms (2.1))
• (Orders of Growth (2.2))
• (Complexity (2.3))
• Number theory (2.4-5)
• (Number theory apps. (2.6))
• (Matrices (2.7))
• Proof strategy (3.1)
• (Sequences (3.2))

• (Summations (3.2))
• (Countability (3.2))
• Inductive Proofs (3.3)
• Recursion (3.4-5)
• Program verification (3.6)
• (Combinatorics (ch. 4))
• (Probability (ch. 5))
• (Recurrences (6.1-3))
• Relations (ch. 7)
• Graph Theory (chs. 89)
• (Boolean Algebra (ch. 10))
• (Computing Theory (ch.11))

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Course Objectives

• Upon completion of this course, the student
  should be able to
• Check validity of simple logical arguments
  (proofs).
• Check the correctness of simple algorithms.
• Recognise and construct simple instances of valid
  logical arguments and correct algorithms.
• Describe the definitions and properties of a
  variety of specific types of discrete structures.
• Correctly read, represent and analyze various
  types of discrete structures using standard
  notations.
• Preparation for CS4026, Formal Models

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Have Fun!

• Many people find Discrete Mathematics more
  enjoyable than, for example, Analysis
• Applicable to just about anything
• Some nice puzzles
• Highly varied