csi2101 discrete structures: introductionlucia/courses/2101-12/lecturenotes/01introduction.pdf ·...
TRANSCRIPT
![Page 1: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/1.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
CSI2101 Discrete Structures: Introduction
Lucia Moura
Winter 2012
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 2: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/2.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Discrete Mathematics is essential to Computer Science!
The various aspects of discrete mathematics form the foundation for:
I modelling computing structuresI designing programs and algorithmsI reasoning about programs and algorithmsI solving real-world problems using the computerI in particular, solving problems in information technology,
i.e. problems involving computers, communication, information.
Aspects of discrete mathematics: (David J. Hunter, Essential of Discrete Mathematics, 2009)
1 logical thinking2 relational thinking3 recursive thinking4 quantitative thinking (counting)5 analytical thinking6 applied thinking
Question: How these 5 aspects appear in the the activities listedabove?
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 3: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/3.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Discrete Mathematics is essential to Computer Science!
The various aspects of discrete mathematics form the foundation for:I modelling computing structures
I designing programs and algorithmsI reasoning about programs and algorithmsI solving real-world problems using the computerI in particular, solving problems in information technology,
i.e. problems involving computers, communication, information.
Aspects of discrete mathematics: (David J. Hunter, Essential of Discrete Mathematics, 2009)
1 logical thinking2 relational thinking3 recursive thinking4 quantitative thinking (counting)5 analytical thinking6 applied thinking
Question: How these 5 aspects appear in the the activities listedabove?
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 4: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/4.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Discrete Mathematics is essential to Computer Science!
The various aspects of discrete mathematics form the foundation for:I modelling computing structuresI designing programs and algorithms
I reasoning about programs and algorithmsI solving real-world problems using the computerI in particular, solving problems in information technology,
i.e. problems involving computers, communication, information.
Aspects of discrete mathematics: (David J. Hunter, Essential of Discrete Mathematics, 2009)
1 logical thinking2 relational thinking3 recursive thinking4 quantitative thinking (counting)5 analytical thinking6 applied thinking
Question: How these 5 aspects appear in the the activities listedabove?
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 5: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/5.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Discrete Mathematics is essential to Computer Science!
The various aspects of discrete mathematics form the foundation for:I modelling computing structuresI designing programs and algorithmsI reasoning about programs and algorithms
I solving real-world problems using the computerI in particular, solving problems in information technology,
i.e. problems involving computers, communication, information.
Aspects of discrete mathematics: (David J. Hunter, Essential of Discrete Mathematics, 2009)
1 logical thinking2 relational thinking3 recursive thinking4 quantitative thinking (counting)5 analytical thinking6 applied thinking
Question: How these 5 aspects appear in the the activities listedabove?
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 6: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/6.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Discrete Mathematics is essential to Computer Science!
The various aspects of discrete mathematics form the foundation for:I modelling computing structuresI designing programs and algorithmsI reasoning about programs and algorithmsI solving real-world problems using the computer
I in particular, solving problems in information technology,i.e. problems involving computers, communication, information.
Aspects of discrete mathematics: (David J. Hunter, Essential of Discrete Mathematics, 2009)
1 logical thinking2 relational thinking3 recursive thinking4 quantitative thinking (counting)5 analytical thinking6 applied thinking
Question: How these 5 aspects appear in the the activities listedabove?
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 7: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/7.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Discrete Mathematics is essential to Computer Science!
The various aspects of discrete mathematics form the foundation for:I modelling computing structuresI designing programs and algorithmsI reasoning about programs and algorithmsI solving real-world problems using the computerI in particular, solving problems in information technology,
i.e. problems involving computers, communication, information.
Aspects of discrete mathematics: (David J. Hunter, Essential of Discrete Mathematics, 2009)
1 logical thinking2 relational thinking3 recursive thinking4 quantitative thinking (counting)5 analytical thinking6 applied thinking
Question: How these 5 aspects appear in the the activities listedabove?
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 8: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/8.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Discrete Mathematics is essential to Computer Science!
The various aspects of discrete mathematics form the foundation for:I modelling computing structuresI designing programs and algorithmsI reasoning about programs and algorithmsI solving real-world problems using the computerI in particular, solving problems in information technology,
i.e. problems involving computers, communication, information.
Aspects of discrete mathematics: (David J. Hunter, Essential of Discrete Mathematics, 2009)
1 logical thinking2 relational thinking3 recursive thinking4 quantitative thinking (counting)5 analytical thinking6 applied thinking
Question: How these 5 aspects appear in the the activities listedabove?
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 9: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/9.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Discrete Mathematics is essential to Computer Science!
The various aspects of discrete mathematics form the foundation for:I modelling computing structuresI designing programs and algorithmsI reasoning about programs and algorithmsI solving real-world problems using the computerI in particular, solving problems in information technology,
i.e. problems involving computers, communication, information.
Aspects of discrete mathematics: (David J. Hunter, Essential of Discrete Mathematics, 2009)
1 logical thinking
2 relational thinking3 recursive thinking4 quantitative thinking (counting)5 analytical thinking6 applied thinking
Question: How these 5 aspects appear in the the activities listedabove?
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 10: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/10.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Discrete Mathematics is essential to Computer Science!
The various aspects of discrete mathematics form the foundation for:I modelling computing structuresI designing programs and algorithmsI reasoning about programs and algorithmsI solving real-world problems using the computerI in particular, solving problems in information technology,
i.e. problems involving computers, communication, information.
Aspects of discrete mathematics: (David J. Hunter, Essential of Discrete Mathematics, 2009)
1 logical thinking2 relational thinking
3 recursive thinking4 quantitative thinking (counting)5 analytical thinking6 applied thinking
Question: How these 5 aspects appear in the the activities listedabove?
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 11: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/11.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Discrete Mathematics is essential to Computer Science!
The various aspects of discrete mathematics form the foundation for:I modelling computing structuresI designing programs and algorithmsI reasoning about programs and algorithmsI solving real-world problems using the computerI in particular, solving problems in information technology,
i.e. problems involving computers, communication, information.
Aspects of discrete mathematics: (David J. Hunter, Essential of Discrete Mathematics, 2009)
1 logical thinking2 relational thinking3 recursive thinking
4 quantitative thinking (counting)5 analytical thinking6 applied thinking
Question: How these 5 aspects appear in the the activities listedabove?
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 12: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/12.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Discrete Mathematics is essential to Computer Science!
The various aspects of discrete mathematics form the foundation for:I modelling computing structuresI designing programs and algorithmsI reasoning about programs and algorithmsI solving real-world problems using the computerI in particular, solving problems in information technology,
i.e. problems involving computers, communication, information.
Aspects of discrete mathematics: (David J. Hunter, Essential of Discrete Mathematics, 2009)
1 logical thinking2 relational thinking3 recursive thinking4 quantitative thinking (counting)
5 analytical thinking6 applied thinking
Question: How these 5 aspects appear in the the activities listedabove?
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 13: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/13.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Discrete Mathematics is essential to Computer Science!
The various aspects of discrete mathematics form the foundation for:I modelling computing structuresI designing programs and algorithmsI reasoning about programs and algorithmsI solving real-world problems using the computerI in particular, solving problems in information technology,
i.e. problems involving computers, communication, information.
Aspects of discrete mathematics: (David J. Hunter, Essential of Discrete Mathematics, 2009)
1 logical thinking2 relational thinking3 recursive thinking4 quantitative thinking (counting)5 analytical thinking
6 applied thinking
Question: How these 5 aspects appear in the the activities listedabove?
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 14: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/14.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Discrete Mathematics is essential to Computer Science!
The various aspects of discrete mathematics form the foundation for:I modelling computing structuresI designing programs and algorithmsI reasoning about programs and algorithmsI solving real-world problems using the computerI in particular, solving problems in information technology,
i.e. problems involving computers, communication, information.
Aspects of discrete mathematics: (David J. Hunter, Essential of Discrete Mathematics, 2009)
1 logical thinking2 relational thinking3 recursive thinking4 quantitative thinking (counting)5 analytical thinking6 applied thinking
Question: How these 5 aspects appear in the the activities listedabove?
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 15: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/15.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Discrete Mathematics is essential to Computer Science!
The various aspects of discrete mathematics form the foundation for:I modelling computing structuresI designing programs and algorithmsI reasoning about programs and algorithmsI solving real-world problems using the computerI in particular, solving problems in information technology,
i.e. problems involving computers, communication, information.
Aspects of discrete mathematics: (David J. Hunter, Essential of Discrete Mathematics, 2009)
1 logical thinking2 relational thinking3 recursive thinking4 quantitative thinking (counting)5 analytical thinking6 applied thinking
Question: How these 5 aspects appear in the the activities listedabove?
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 16: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/16.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Logical Thinking
formal logic(symbolic manipulation of notation; logical not-thinking :-)))) )
propositional logicpropositional calculus allows us to make logical deductions formally
predicate logicmake a proposition to depend on a variable and we get a predicate;here the logical deductions include quantifiers (for all, there exists) infront of the predicates)
methods of proof: direct, by contraposition, by contradictionuse what you learned in formal/symbolic logic, to guide yourreasoning on mathematical proofs (written in paragraph form)
logic in programmingimperative programming: conditional statements (if-then-else, do-while)
logic programming languages (e.g. prolog): uses the rules of predicate logic
logic in circuits
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 17: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/17.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Logical Thinking
formal logic(symbolic manipulation of notation; logical not-thinking :-)))) )
propositional logicpropositional calculus allows us to make logical deductions formally
predicate logicmake a proposition to depend on a variable and we get a predicate;here the logical deductions include quantifiers (for all, there exists) infront of the predicates)
methods of proof: direct, by contraposition, by contradictionuse what you learned in formal/symbolic logic, to guide yourreasoning on mathematical proofs (written in paragraph form)
logic in programmingimperative programming: conditional statements (if-then-else, do-while)
logic programming languages (e.g. prolog): uses the rules of predicate logic
logic in circuits
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 18: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/18.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Logical Thinking
formal logic(symbolic manipulation of notation; logical not-thinking :-)))) )
propositional logicpropositional calculus allows us to make logical deductions formally
predicate logicmake a proposition to depend on a variable and we get a predicate;here the logical deductions include quantifiers (for all, there exists) infront of the predicates)
methods of proof: direct, by contraposition, by contradictionuse what you learned in formal/symbolic logic, to guide yourreasoning on mathematical proofs (written in paragraph form)
logic in programmingimperative programming: conditional statements (if-then-else, do-while)
logic programming languages (e.g. prolog): uses the rules of predicate logic
logic in circuits
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 19: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/19.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Logical Thinking
formal logic(symbolic manipulation of notation; logical not-thinking :-)))) )
propositional logicpropositional calculus allows us to make logical deductions formally
predicate logicmake a proposition to depend on a variable and we get a predicate;here the logical deductions include quantifiers (for all, there exists) infront of the predicates)
methods of proof: direct, by contraposition, by contradictionuse what you learned in formal/symbolic logic, to guide yourreasoning on mathematical proofs (written in paragraph form)
logic in programmingimperative programming: conditional statements (if-then-else, do-while)
logic programming languages (e.g. prolog): uses the rules of predicate logic
logic in circuits
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 20: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/20.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Logical Thinking
formal logic(symbolic manipulation of notation; logical not-thinking :-)))) )
propositional logicpropositional calculus allows us to make logical deductions formally
predicate logicmake a proposition to depend on a variable and we get a predicate;here the logical deductions include quantifiers (for all, there exists) infront of the predicates)
methods of proof: direct, by contraposition, by contradictionuse what you learned in formal/symbolic logic, to guide yourreasoning on mathematical proofs (written in paragraph form)
logic in programmingimperative programming: conditional statements (if-then-else, do-while)
logic programming languages (e.g. prolog): uses the rules of predicate logic
logic in circuits
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 21: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/21.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Logical Thinking
formal logic(symbolic manipulation of notation; logical not-thinking :-)))) )
propositional logicpropositional calculus allows us to make logical deductions formally
predicate logicmake a proposition to depend on a variable and we get a predicate;here the logical deductions include quantifiers (for all, there exists) infront of the predicates)
methods of proof: direct, by contraposition, by contradictionuse what you learned in formal/symbolic logic, to guide yourreasoning on mathematical proofs (written in paragraph form)
logic in programmingimperative programming: conditional statements (if-then-else, do-while)
logic programming languages (e.g. prolog): uses the rules of predicate logic
logic in circuits
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 22: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/22.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Relational Thinking
It deals with the following type of structures:
I setsI functionsI relationsI partial orderingsI graph theory
Question 1: what are each of these structures?
Question 2: give examples of situations where they can be applied incomputer science.
I Databases: table=relation; record=n-ary tupleI Dependency of task executions (partial ordering);
topological sorting: order tasks respecting dependencies.I Graphs: networks (communication, roads, social),
conflicts (timetabling, coloring maps),hierarquies (rooted trees), diagrams (binary relations).
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 23: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/23.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Relational Thinking
It deals with the following type of structures:
I sets
I functionsI relationsI partial orderingsI graph theory
Question 1: what are each of these structures?
Question 2: give examples of situations where they can be applied incomputer science.
I Databases: table=relation; record=n-ary tupleI Dependency of task executions (partial ordering);
topological sorting: order tasks respecting dependencies.I Graphs: networks (communication, roads, social),
conflicts (timetabling, coloring maps),hierarquies (rooted trees), diagrams (binary relations).
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 24: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/24.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Relational Thinking
It deals with the following type of structures:
I setsI functions
I relationsI partial orderingsI graph theory
Question 1: what are each of these structures?
Question 2: give examples of situations where they can be applied incomputer science.
I Databases: table=relation; record=n-ary tupleI Dependency of task executions (partial ordering);
topological sorting: order tasks respecting dependencies.I Graphs: networks (communication, roads, social),
conflicts (timetabling, coloring maps),hierarquies (rooted trees), diagrams (binary relations).
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 25: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/25.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Relational Thinking
It deals with the following type of structures:
I setsI functionsI relations
I partial orderingsI graph theory
Question 1: what are each of these structures?
Question 2: give examples of situations where they can be applied incomputer science.
I Databases: table=relation; record=n-ary tupleI Dependency of task executions (partial ordering);
topological sorting: order tasks respecting dependencies.I Graphs: networks (communication, roads, social),
conflicts (timetabling, coloring maps),hierarquies (rooted trees), diagrams (binary relations).
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 26: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/26.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Relational Thinking
It deals with the following type of structures:
I setsI functionsI relationsI partial orderings
I graph theory
Question 1: what are each of these structures?
Question 2: give examples of situations where they can be applied incomputer science.
I Databases: table=relation; record=n-ary tupleI Dependency of task executions (partial ordering);
topological sorting: order tasks respecting dependencies.I Graphs: networks (communication, roads, social),
conflicts (timetabling, coloring maps),hierarquies (rooted trees), diagrams (binary relations).
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 27: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/27.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Relational Thinking
It deals with the following type of structures:
I setsI functionsI relationsI partial orderingsI graph theory
Question 1: what are each of these structures?
Question 2: give examples of situations where they can be applied incomputer science.
I Databases: table=relation; record=n-ary tupleI Dependency of task executions (partial ordering);
topological sorting: order tasks respecting dependencies.I Graphs: networks (communication, roads, social),
conflicts (timetabling, coloring maps),hierarquies (rooted trees), diagrams (binary relations).
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 28: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/28.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Relational Thinking
It deals with the following type of structures:
I setsI functionsI relationsI partial orderingsI graph theory
Question 1: what are each of these structures?
Question 2: give examples of situations where they can be applied incomputer science.
I Databases: table=relation; record=n-ary tupleI Dependency of task executions (partial ordering);
topological sorting: order tasks respecting dependencies.I Graphs: networks (communication, roads, social),
conflicts (timetabling, coloring maps),hierarquies (rooted trees), diagrams (binary relations).
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 29: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/29.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Relational Thinking
It deals with the following type of structures:
I setsI functionsI relationsI partial orderingsI graph theory
Question 1: what are each of these structures?
Question 2: give examples of situations where they can be applied incomputer science.
I Databases: table=relation; record=n-ary tupleI Dependency of task executions (partial ordering);
topological sorting: order tasks respecting dependencies.I Graphs: networks (communication, roads, social),
conflicts (timetabling, coloring maps),hierarquies (rooted trees), diagrams (binary relations).
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 30: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/30.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Relational Thinking
It deals with the following type of structures:
I setsI functionsI relationsI partial orderingsI graph theory
Question 1: what are each of these structures?
Question 2: give examples of situations where they can be applied incomputer science.
I Databases: table=relation; record=n-ary tuple
I Dependency of task executions (partial ordering);topological sorting: order tasks respecting dependencies.
I Graphs: networks (communication, roads, social),conflicts (timetabling, coloring maps),hierarquies (rooted trees), diagrams (binary relations).
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 31: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/31.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Relational Thinking
It deals with the following type of structures:
I setsI functionsI relationsI partial orderingsI graph theory
Question 1: what are each of these structures?
Question 2: give examples of situations where they can be applied incomputer science.
I Databases: table=relation; record=n-ary tupleI Dependency of task executions (partial ordering);
topological sorting: order tasks respecting dependencies.
I Graphs: networks (communication, roads, social),conflicts (timetabling, coloring maps),hierarquies (rooted trees), diagrams (binary relations).
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 32: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/32.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Relational Thinking
It deals with the following type of structures:
I setsI functionsI relationsI partial orderingsI graph theory
Question 1: what are each of these structures?
Question 2: give examples of situations where they can be applied incomputer science.
I Databases: table=relation; record=n-ary tupleI Dependency of task executions (partial ordering);
topological sorting: order tasks respecting dependencies.I Graphs: networks (communication, roads, social),
conflicts (timetabling, coloring maps),hierarquies (rooted trees), diagrams (binary relations).
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 33: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/33.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Recursive Thinking
Recurrence relationsRecursively defined sequences of numbers. e.g. Fibonacci sequence.
Recursive definitionse.g. binary trees, recursive geometry/fractals
Proofs by inductionProve that P (n) is true for all n ≥ 0:basis: P (0) is true + induction step P (n)⇒ P (n + 1)Recursive data structurese.g. binary search trees
Recursive algorithmse.g. binary search, mergesort, solving towers of Hanoi.
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 34: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/34.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Recursive Thinking
Recurrence relationsRecursively defined sequences of numbers. e.g. Fibonacci sequence.
Recursive definitionse.g. binary trees, recursive geometry/fractals
Proofs by inductionProve that P (n) is true for all n ≥ 0:basis: P (0) is true + induction step P (n)⇒ P (n + 1)Recursive data structurese.g. binary search trees
Recursive algorithmse.g. binary search, mergesort, solving towers of Hanoi.
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 35: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/35.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Recursive Thinking
Recurrence relationsRecursively defined sequences of numbers. e.g. Fibonacci sequence.
Recursive definitionse.g. binary trees, recursive geometry/fractals
Proofs by inductionProve that P (n) is true for all n ≥ 0:basis: P (0) is true + induction step P (n)⇒ P (n + 1)
Recursive data structurese.g. binary search trees
Recursive algorithmse.g. binary search, mergesort, solving towers of Hanoi.
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 36: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/36.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Recursive Thinking
Recurrence relationsRecursively defined sequences of numbers. e.g. Fibonacci sequence.
Recursive definitionse.g. binary trees, recursive geometry/fractals
Proofs by inductionProve that P (n) is true for all n ≥ 0:basis: P (0) is true + induction step P (n)⇒ P (n + 1)Recursive data structurese.g. binary search trees
Recursive algorithmse.g. binary search, mergesort, solving towers of Hanoi.
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 37: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/37.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Recursive Thinking
Recurrence relationsRecursively defined sequences of numbers. e.g. Fibonacci sequence.
Recursive definitionse.g. binary trees, recursive geometry/fractals
Proofs by inductionProve that P (n) is true for all n ≥ 0:basis: P (0) is true + induction step P (n)⇒ P (n + 1)Recursive data structurese.g. binary search trees
Recursive algorithmse.g. binary search, mergesort, solving towers of Hanoi.
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 38: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/38.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Quantitative Thinking
counting,
combinations, permutations, arrangements,
the pigeonhole principle,
discrete probability,
counting operations in algorithms,
estimating growth of functions, big-Oh notation.
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 39: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/39.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Quantitative Thinking
counting,
combinations, permutations, arrangements,
the pigeonhole principle,
discrete probability,
counting operations in algorithms,
estimating growth of functions, big-Oh notation.
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 40: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/40.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Quantitative Thinking
counting,
combinations, permutations, arrangements,
the pigeonhole principle,
discrete probability,
counting operations in algorithms,
estimating growth of functions, big-Oh notation.
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 41: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/41.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Quantitative Thinking
counting,
combinations, permutations, arrangements,
the pigeonhole principle,
discrete probability,
counting operations in algorithms,
estimating growth of functions, big-Oh notation.
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 42: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/42.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Quantitative Thinking
counting,
combinations, permutations, arrangements,
the pigeonhole principle,
discrete probability,
counting operations in algorithms,
estimating growth of functions, big-Oh notation.
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 43: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/43.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Quantitative Thinking
counting,
combinations, permutations, arrangements,
the pigeonhole principle,
discrete probability,
counting operations in algorithms,
estimating growth of functions, big-Oh notation.
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 44: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/44.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Analytical Thinking
Apply previous tools to analyze problems of interest such as:
I Writing programs that are correct.software engineering tools: testing versus verification.
F loop invariants,F program correctness and verification.
I Writing algorithms that are efficient.
F algorithm complexity,F analysis of algorithms.
Question: How previous tools can be applied in each of the aboveareas?
I This question will be answered more fully by the studies in this course.
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 45: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/45.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Analytical Thinking
Apply previous tools to analyze problems of interest such as:
I Writing programs that are correct.software engineering tools: testing versus verification.
F loop invariants,F program correctness and verification.
I Writing algorithms that are efficient.
F algorithm complexity,F analysis of algorithms.
Question: How previous tools can be applied in each of the aboveareas?
I This question will be answered more fully by the studies in this course.
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 46: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/46.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Analytical Thinking
Apply previous tools to analyze problems of interest such as:
I Writing programs that are correct.software engineering tools: testing versus verification.
F loop invariants,
F program correctness and verification.
I Writing algorithms that are efficient.
F algorithm complexity,F analysis of algorithms.
Question: How previous tools can be applied in each of the aboveareas?
I This question will be answered more fully by the studies in this course.
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 47: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/47.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Analytical Thinking
Apply previous tools to analyze problems of interest such as:
I Writing programs that are correct.software engineering tools: testing versus verification.
F loop invariants,F program correctness and verification.
I Writing algorithms that are efficient.
F algorithm complexity,F analysis of algorithms.
Question: How previous tools can be applied in each of the aboveareas?
I This question will be answered more fully by the studies in this course.
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 48: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/48.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Analytical Thinking
Apply previous tools to analyze problems of interest such as:
I Writing programs that are correct.software engineering tools: testing versus verification.
F loop invariants,F program correctness and verification.
I Writing algorithms that are efficient.
F algorithm complexity,F analysis of algorithms.
Question: How previous tools can be applied in each of the aboveareas?
I This question will be answered more fully by the studies in this course.
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 49: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/49.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Analytical Thinking
Apply previous tools to analyze problems of interest such as:
I Writing programs that are correct.software engineering tools: testing versus verification.
F loop invariants,F program correctness and verification.
I Writing algorithms that are efficient.F algorithm complexity,
F analysis of algorithms.
Question: How previous tools can be applied in each of the aboveareas?
I This question will be answered more fully by the studies in this course.
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 50: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/50.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Analytical Thinking
Apply previous tools to analyze problems of interest such as:
I Writing programs that are correct.software engineering tools: testing versus verification.
F loop invariants,F program correctness and verification.
I Writing algorithms that are efficient.F algorithm complexity,F analysis of algorithms.
Question: How previous tools can be applied in each of the aboveareas?
I This question will be answered more fully by the studies in this course.
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 51: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/51.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Analytical Thinking
Apply previous tools to analyze problems of interest such as:
I Writing programs that are correct.software engineering tools: testing versus verification.
F loop invariants,F program correctness and verification.
I Writing algorithms that are efficient.F algorithm complexity,F analysis of algorithms.
Question: How previous tools can be applied in each of the aboveareas?
I This question will be answered more fully by the studies in this course.
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 52: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/52.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Analytical Thinking
Apply previous tools to analyze problems of interest such as:
I Writing programs that are correct.software engineering tools: testing versus verification.
F loop invariants,F program correctness and verification.
I Writing algorithms that are efficient.F algorithm complexity,F analysis of algorithms.
Question: How previous tools can be applied in each of the aboveareas?
I This question will be answered more fully by the studies in this course.
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 53: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/53.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Applied Thinking
Making the bridge between the mathematical tools and problems weneed to solve.
I problem solving skillsI modelling
Before using tools we need to learn the language and methods.
A lot of the course will focus on acquiring the mathematical skills.But we don’t want to lose sight of their use in applications.
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 54: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/54.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Applied Thinking
Making the bridge between the mathematical tools and problems weneed to solve.
I problem solving skills
I modelling
Before using tools we need to learn the language and methods.
A lot of the course will focus on acquiring the mathematical skills.But we don’t want to lose sight of their use in applications.
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 55: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/55.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Applied Thinking
Making the bridge between the mathematical tools and problems weneed to solve.
I problem solving skillsI modelling
Before using tools we need to learn the language and methods.
A lot of the course will focus on acquiring the mathematical skills.But we don’t want to lose sight of their use in applications.
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 56: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/56.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Applied Thinking
Making the bridge between the mathematical tools and problems weneed to solve.
I problem solving skillsI modelling
Before using tools we need to learn the language and methods.
A lot of the course will focus on acquiring the mathematical skills.But we don’t want to lose sight of their use in applications.
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 57: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/57.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Why Discrete Mathematics ?
Applied Thinking
Making the bridge between the mathematical tools and problems weneed to solve.
I problem solving skillsI modelling
Before using tools we need to learn the language and methods.
A lot of the course will focus on acquiring the mathematical skills.But we don’t want to lose sight of their use in applications.
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 58: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/58.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Course Content Overview
Calendar description:
CSI2101 Discrete Structures (3,1.5,0) 3 cr. Discrete structures as theyapply to computer science, algorithm analysis and design. Predicate logic.Review of proof techniques; application of induction to computingproblems. Graph theory applications in information technology. Programcorrectness, preconditions, postconditions and invariants. Analysis ofrecursive programs using recurrence relations. Properties of integers andbasic cryptographical applications. Prerequisite: MAT1348.
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 59: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/59.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Course Content Overview
Objectives:
Discrete mathematics form the foundation for computer science;it is essential in every branch of computing.
In MAT1348 (discrete mathematics for computing) you have beenintroduced to fundamental problems and objects in discretemathematics.
In CSI2101 (discrete structures) you will learn:I more advanced concepts in discrete mathematicsI more problem solving, modelling, logical reasoning and writing precise
proofsI how to apply concepts to various types of problems in computing:
analyse an algorithm, prove the correctness of a program, model anetwork problem with graphs, use number theory in cryptography, etc.
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 60: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/60.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Course Content Overview
Textbook
References:
Kenneth H. Rosen, Discrete Mathematics and Its Applications,Seventh Edition, McGraw Hill, 2012.(same textbook as normally used for MAT1348; we will use differentsections!)
CSI2101 Discrete Structures, Winter 2012 Lucia Moura
![Page 61: CSI2101 Discrete Structures: Introductionlucia/courses/2101-12/lecturenotes/01Introduction.pdf · CSI2101 Discrete Structures: Introduction Lucia Moura Winter 2012 CSI2101 Discrete](https://reader034.vdocument.in/reader034/viewer/2022042101/5e7dfbe49637a32132315955/html5/thumbnails/61.jpg)
Discrete Mathematics and Discrete Structures Course Content Overview
Course Content Overview
Topic by topic outline: (approximate number of lectures, order may vary)
1 Introduction (1)
2 Propositionl logic (1)
3 Predicate logic (3)
4 Rules of inference/proof methods (2)
5 Basic number theory and applications (4)
6 Induction and applications. (4)Program correctness and verification (1)
7 Solving recurrence relations. Complexity of divide-and-conqueralgorithms. (4)
8 Graphs (3)
CSI2101 Discrete Structures, Winter 2012 Lucia Moura