Outline Course Info What is Discrete Mathematics?
Discrete Structures of Computer ScienceIntroduction
Gazihan Alankus(Based on original slides by Brahim Hnich)
July 23, 2012
Outline Course Info What is Discrete Mathematics?
1 Course Info
2 What is Discrete Mathematics?Continuous MathematicsDiscrete Mathematics
Outline Course Info What is Discrete Mathematics?
Course Info
Instructors:
Asst. Prof. Dr. Gazihan AlankusTeaching assistant: NoneCourse Webpage
http://homes.ieu.edu.tr/∼galankus
Outline Course Info What is Discrete Mathematics?
Course Info
Instructors:Asst. Prof. Dr. Gazihan Alankus
Teaching assistant: NoneCourse Webpage
http://homes.ieu.edu.tr/∼galankus
Outline Course Info What is Discrete Mathematics?
Course Info
Instructors:Asst. Prof. Dr. Gazihan Alankus
Teaching assistant: None
Course Webpage
http://homes.ieu.edu.tr/∼galankus
Outline Course Info What is Discrete Mathematics?
Course Info
Instructors:Asst. Prof. Dr. Gazihan Alankus
Teaching assistant: NoneCourse Webpage
http://homes.ieu.edu.tr/∼galankus
Outline Course Info What is Discrete Mathematics?
Course Info
Course objective
Introduce basic concepts and applications of discretemathematics
Syllabus
Logic and formal reasoningSet theoryRelations and functionsMathematical inductionCombinatoricsFinite probabilityGraphs and Trees
Outline Course Info What is Discrete Mathematics?
Course Info
Course objectiveIntroduce basic concepts and applications of discretemathematics
Syllabus
Logic and formal reasoningSet theoryRelations and functionsMathematical inductionCombinatoricsFinite probabilityGraphs and Trees
Outline Course Info What is Discrete Mathematics?
Course Info
Course objectiveIntroduce basic concepts and applications of discretemathematics
Syllabus
Logic and formal reasoningSet theoryRelations and functionsMathematical inductionCombinatoricsFinite probabilityGraphs and Trees
Outline Course Info What is Discrete Mathematics?
Course Info
Course objectiveIntroduce basic concepts and applications of discretemathematics
SyllabusLogic and formal reasoning
Set theoryRelations and functionsMathematical inductionCombinatoricsFinite probabilityGraphs and Trees
Outline Course Info What is Discrete Mathematics?
Course Info
Course objectiveIntroduce basic concepts and applications of discretemathematics
SyllabusLogic and formal reasoningSet theory
Relations and functionsMathematical inductionCombinatoricsFinite probabilityGraphs and Trees
Outline Course Info What is Discrete Mathematics?
Course Info
Course objectiveIntroduce basic concepts and applications of discretemathematics
SyllabusLogic and formal reasoningSet theoryRelations and functions
Mathematical inductionCombinatoricsFinite probabilityGraphs and Trees
Outline Course Info What is Discrete Mathematics?
Course Info
Course objectiveIntroduce basic concepts and applications of discretemathematics
SyllabusLogic and formal reasoningSet theoryRelations and functionsMathematical induction
CombinatoricsFinite probabilityGraphs and Trees
Outline Course Info What is Discrete Mathematics?
Course Info
Course objectiveIntroduce basic concepts and applications of discretemathematics
SyllabusLogic and formal reasoningSet theoryRelations and functionsMathematical inductionCombinatorics
Finite probabilityGraphs and Trees
Outline Course Info What is Discrete Mathematics?
Course Info
Course objectiveIntroduce basic concepts and applications of discretemathematics
SyllabusLogic and formal reasoningSet theoryRelations and functionsMathematical inductionCombinatoricsFinite probability
Graphs and Trees
Outline Course Info What is Discrete Mathematics?
Course Info
Course objectiveIntroduce basic concepts and applications of discretemathematics
SyllabusLogic and formal reasoningSet theoryRelations and functionsMathematical inductionCombinatoricsFinite probabilityGraphs and Trees
Outline Course Info What is Discrete Mathematics?
Textbook
Discrete and Combinatorial Mathematics (An AppliedIntroduction) by Ralph. G. Grimaldi, Addison-Wesley: 5th
Edition
We will not cover all the chapters of the bookWe will not cover all sections of all covered chaptersWe will not follow the order of the bookIn few cases the notation and terminology will be differentfrom that of the bookThe contents uncovered in slides/lectures are not required
Outline Course Info What is Discrete Mathematics?
Textbook
Discrete and Combinatorial Mathematics (An AppliedIntroduction) by Ralph. G. Grimaldi, Addison-Wesley: 5th
EditionWe will not cover all the chapters of the book
We will not cover all sections of all covered chaptersWe will not follow the order of the bookIn few cases the notation and terminology will be differentfrom that of the bookThe contents uncovered in slides/lectures are not required
Outline Course Info What is Discrete Mathematics?
Textbook
Discrete and Combinatorial Mathematics (An AppliedIntroduction) by Ralph. G. Grimaldi, Addison-Wesley: 5th
EditionWe will not cover all the chapters of the bookWe will not cover all sections of all covered chapters
We will not follow the order of the bookIn few cases the notation and terminology will be differentfrom that of the bookThe contents uncovered in slides/lectures are not required
Outline Course Info What is Discrete Mathematics?
Textbook
Discrete and Combinatorial Mathematics (An AppliedIntroduction) by Ralph. G. Grimaldi, Addison-Wesley: 5th
EditionWe will not cover all the chapters of the bookWe will not cover all sections of all covered chaptersWe will not follow the order of the book
In few cases the notation and terminology will be differentfrom that of the bookThe contents uncovered in slides/lectures are not required
Outline Course Info What is Discrete Mathematics?
Textbook
Discrete and Combinatorial Mathematics (An AppliedIntroduction) by Ralph. G. Grimaldi, Addison-Wesley: 5th
EditionWe will not cover all the chapters of the bookWe will not cover all sections of all covered chaptersWe will not follow the order of the bookIn few cases the notation and terminology will be differentfrom that of the book
The contents uncovered in slides/lectures are not required
Outline Course Info What is Discrete Mathematics?
Textbook
Discrete and Combinatorial Mathematics (An AppliedIntroduction) by Ralph. G. Grimaldi, Addison-Wesley: 5th
EditionWe will not cover all the chapters of the bookWe will not cover all sections of all covered chaptersWe will not follow the order of the bookIn few cases the notation and terminology will be differentfrom that of the bookThe contents uncovered in slides/lectures are not required
Outline Course Info What is Discrete Mathematics?
Grading
6 Homeworks each worth 5%
1 Midterm exam worth 30%
Midterm will be on August 23
1 Final exam worth 40%
Outline Course Info What is Discrete Mathematics?
Grading
6 Homeworks each worth 5%1 Midterm exam worth 30%
Midterm will be on August 23
1 Final exam worth 40%
Outline Course Info What is Discrete Mathematics?
Grading
6 Homeworks each worth 5%1 Midterm exam worth 30%
Midterm will be on August 231 Final exam worth 40%
Outline Course Info What is Discrete Mathematics?
Two Mathematics
Continuous mathematics
Discrete mathematics
Outline Course Info What is Discrete Mathematics?
Two Mathematics
Continuous mathematicsDiscrete mathematics
Outline Course Info What is Discrete Mathematics?
Calculus
Calculus (Latin, calculus, a small stone used for counting)
It is a branch of mathematicsIt includes the study of limits, derivatives, integrals, andinfinite series
Outline Course Info What is Discrete Mathematics?
Calculus
Calculus (Latin, calculus, a small stone used for counting)It is a branch of mathematics
It includes the study of limits, derivatives, integrals, andinfinite series
Outline Course Info What is Discrete Mathematics?
Calculus
Calculus (Latin, calculus, a small stone used for counting)It is a branch of mathematicsIt includes the study of limits, derivatives, integrals, andinfinite series
Outline Course Info What is Discrete Mathematics?
Trigonometry
Trigonometry (from Greek trigo-non ”triangle” + metron”measure”)
It is a branch of mathematics that deals with triangles,It deals with relationships between the sides and the angles oftrianglesIt also deals with the trigonometric functions, which describethose relationships.
Outline Course Info What is Discrete Mathematics?
Trigonometry
Trigonometry (from Greek trigo-non ”triangle” + metron”measure”)It is a branch of mathematics that deals with triangles,
It deals with relationships between the sides and the angles oftrianglesIt also deals with the trigonometric functions, which describethose relationships.
Outline Course Info What is Discrete Mathematics?
Trigonometry
Trigonometry (from Greek trigo-non ”triangle” + metron”measure”)It is a branch of mathematics that deals with triangles,It deals with relationships between the sides and the angles oftriangles
It also deals with the trigonometric functions, which describethose relationships.
Outline Course Info What is Discrete Mathematics?
Trigonometry
Trigonometry (from Greek trigo-non ”triangle” + metron”measure”)It is a branch of mathematics that deals with triangles,It deals with relationships between the sides and the angles oftrianglesIt also deals with the trigonometric functions, which describethose relationships.
Outline Course Info What is Discrete Mathematics?
Dynamic Systems
The dynamical system concept is a mathematicalformalization for any fixed ”rule” which describes the timedependence of a point’s position in its ambient space
Examples include the mathematical models that describe
1 the swinging of a clock pendulum2 the flow of water in a pipe3 the number of fish each spring in a lake
Outline Course Info What is Discrete Mathematics?
Dynamic Systems
The dynamical system concept is a mathematicalformalization for any fixed ”rule” which describes the timedependence of a point’s position in its ambient spaceExamples include the mathematical models that describe
1 the swinging of a clock pendulum2 the flow of water in a pipe3 the number of fish each spring in a lake
Outline Course Info What is Discrete Mathematics?
Dynamic Systems
The dynamical system concept is a mathematicalformalization for any fixed ”rule” which describes the timedependence of a point’s position in its ambient spaceExamples include the mathematical models that describe
1 the swinging of a clock pendulum
2 the flow of water in a pipe3 the number of fish each spring in a lake
Outline Course Info What is Discrete Mathematics?
Dynamic Systems
The dynamical system concept is a mathematicalformalization for any fixed ”rule” which describes the timedependence of a point’s position in its ambient spaceExamples include the mathematical models that describe
1 the swinging of a clock pendulum2 the flow of water in a pipe
3 the number of fish each spring in a lake
Outline Course Info What is Discrete Mathematics?
Dynamic Systems
The dynamical system concept is a mathematicalformalization for any fixed ”rule” which describes the timedependence of a point’s position in its ambient spaceExamples include the mathematical models that describe
1 the swinging of a clock pendulum2 the flow of water in a pipe3 the number of fish each spring in a lake
Outline Course Info What is Discrete Mathematics?
Differential Equations
A differential equation is a mathematical equation for anunknown function of one or several variables that relates thevalues of the function itself and of its derivatives of variousorders
They play a prominent role in engineering, physics, economicsand other disciplines
Outline Course Info What is Discrete Mathematics?
Differential Equations
A differential equation is a mathematical equation for anunknown function of one or several variables that relates thevalues of the function itself and of its derivatives of variousordersThey play a prominent role in engineering, physics, economicsand other disciplines
Outline Course Info What is Discrete Mathematics?
Chaos Theory
Chaos theory describes the behavior of certain systems whosestate evolves with time
These systems may exhibit dynamics that are highly sensitiveto initial conditions (popularly referred to as the butterflyeffect)
Outline Course Info What is Discrete Mathematics?
Chaos Theory
Chaos theory describes the behavior of certain systems whosestate evolves with timeThese systems may exhibit dynamics that are highly sensitiveto initial conditions (popularly referred to as the butterflyeffect)
Outline Course Info What is Discrete Mathematics?
Logic and Formal Reasoning
Logic is the study of the principles of valid demonstration andinference
Logic investigates and classifies the structure of statementsand arguments
through the study of formal systems of inference and the studyof arguments in natural language
Outline Course Info What is Discrete Mathematics?
Logic and Formal Reasoning
Logic is the study of the principles of valid demonstration andinferenceLogic investigates and classifies the structure of statementsand arguments
through the study of formal systems of inference and the studyof arguments in natural language
Outline Course Info What is Discrete Mathematics?
Logic and Formal Reasoning
Logic is the study of the principles of valid demonstration andinferenceLogic investigates and classifies the structure of statementsand arguments
through the study of formal systems of inference and the studyof arguments in natural language
Outline Course Info What is Discrete Mathematics?
Set Theory
Set theory is the branch of mathematics that studies sets(collections of objects)
It is the most common foundational system for mathematicsThe language of set theory is used in the definitions of nearlyall mathematical objects
Outline Course Info What is Discrete Mathematics?
Set Theory
Set theory is the branch of mathematics that studies sets(collections of objects)It is the most common foundational system for mathematics
The language of set theory is used in the definitions of nearlyall mathematical objects
Outline Course Info What is Discrete Mathematics?
Set Theory
Set theory is the branch of mathematics that studies sets(collections of objects)It is the most common foundational system for mathematicsThe language of set theory is used in the definitions of nearlyall mathematical objects
Outline Course Info What is Discrete Mathematics?
Combinatorics
Combinatorics is a branch of pure mathematics concerning thestudy of discrete (and usually finite) objects
Outline Course Info What is Discrete Mathematics?
Combinatorics
Aspects of combinatorics include
”counting” the objects satisfying certain criteria (enumerativecombinatorics)deciding when the criteria can be metconstructing and analyzing objects meeting the criteria (as incombinatorial designs)finding ”largest”, ”smallest”, or ”optimal” objects(combinatorial optimization)finding algebraic structures these objects may have (algebraiccombinatorics)
Outline Course Info What is Discrete Mathematics?
Combinatorics
Aspects of combinatorics include”counting” the objects satisfying certain criteria (enumerativecombinatorics)
deciding when the criteria can be metconstructing and analyzing objects meeting the criteria (as incombinatorial designs)finding ”largest”, ”smallest”, or ”optimal” objects(combinatorial optimization)finding algebraic structures these objects may have (algebraiccombinatorics)
Outline Course Info What is Discrete Mathematics?
Combinatorics
Aspects of combinatorics include”counting” the objects satisfying certain criteria (enumerativecombinatorics)deciding when the criteria can be met
constructing and analyzing objects meeting the criteria (as incombinatorial designs)finding ”largest”, ”smallest”, or ”optimal” objects(combinatorial optimization)finding algebraic structures these objects may have (algebraiccombinatorics)
Outline Course Info What is Discrete Mathematics?
Combinatorics
Aspects of combinatorics include”counting” the objects satisfying certain criteria (enumerativecombinatorics)deciding when the criteria can be metconstructing and analyzing objects meeting the criteria (as incombinatorial designs)
finding ”largest”, ”smallest”, or ”optimal” objects(combinatorial optimization)finding algebraic structures these objects may have (algebraiccombinatorics)
Outline Course Info What is Discrete Mathematics?
Combinatorics
Aspects of combinatorics include”counting” the objects satisfying certain criteria (enumerativecombinatorics)deciding when the criteria can be metconstructing and analyzing objects meeting the criteria (as incombinatorial designs)finding ”largest”, ”smallest”, or ”optimal” objects(combinatorial optimization)
finding algebraic structures these objects may have (algebraiccombinatorics)
Outline Course Info What is Discrete Mathematics?
Combinatorics
Aspects of combinatorics include”counting” the objects satisfying certain criteria (enumerativecombinatorics)deciding when the criteria can be metconstructing and analyzing objects meeting the criteria (as incombinatorial designs)finding ”largest”, ”smallest”, or ”optimal” objects(combinatorial optimization)finding algebraic structures these objects may have (algebraiccombinatorics)
Outline Course Info What is Discrete Mathematics?
Theory of Computation
Is the branch of computer science that deals with whether andhow efficiently problems can be solved on a model ofcomputation, using an algorithmThe field is divided into two major branches: computabilitytheory and complexity theory
Outline Course Info What is Discrete Mathematics?
Graph Theory
Graph theory is the study of graphsGraphs are mathematical structures used to model pairwiserelations between objects from a certain collection
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