Discrete lecture 01
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Transcript of Discrete lecture 01
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CS335- Discrete Mathematics
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Agenda
Course policies
Quick Overview
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Grading Scheme
Quize + Assignments 12 Midterm 18 Final 30
Total 60
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Recommended Books
“Discrete Mathematics with Application” by Susana.
K.H. Rosen, Discrete Mathematics and its Applications, (5th Edition), McGraw Hill 1999.
“Discrete Mathematical Structures” by B. Kalman Prentice Hall (1996).
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Grading (cont’d)
Exams/Quizzes can be from the following:
Current lecture Material covered in any previous lecture Reading assignments From any assigned homework.
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Academic Dishonesty
Any form of cheating on exams/assignments/quizzes is subject to a penalty.
Assignment Copy may lead to zero in all assignments.
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Introduction
Discrete mathematics describes processes that consist of a sequence of individual steps.
This contrasts with calculus, which describes processes that change in a continuous fashion.
Whereas the ideas of calculus were fundamental to the science and technology of the industrial revolution, the ideas of discrete mathematics underlie the science and technology of the computer age
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Discrete Continuous – 5a
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What Is Discrete Mathematics?
Definition Discrete Mathematics Discrete Mathematics is a collection of
mathematical topics that examine and use finite or countably infinite mathematical objects.
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1010
Quick Overview - Topics
Logic and Sets Make notions you’re already used to from
programming a little more rigorous (operators) Fundamental to all mathematical disciplines Useful for digital circuits, hardware design
Elementary Number Theory Get to rediscover the old reliable number and
find out some surprising facts Very useful in crypto-systems
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1111
Quick Overview - Topics
Proofs (especially induction) If you want to debug a program beyond a doubt,
prove that it’s bug-free Proof-theory has recently also been shown to be
useful in discovering bugs in pre-production hardware
Counting and Combinatorics Compute your odds of winning lottery Important for predicting how long certain
computer program will take to finish Useful in designing algorithms
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1212
Quick Overview - Topics
Graph Theory Many clever data-structures for organizing information
and making programs highly efficient are based on graph theory
Very useful in describing problems in Databases Operating Systems Networks EVERY CS DISCIPLINE!!!!
Trees Data structures for organizing information and making
programs efficient
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CS-708 13
What is Discrete Mathematics – 6
Set of Integers & Real Numbers – 5b
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CS-708 14
Logic – 7
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CS-708 15
Statement – 8a
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CS-708 16
Examples – 8b
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CS-708 17
Truth Values of Propositions – 8c
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CS-708 18
Examples – 9a
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CS-708 19
Statements & Truth Values – 9b
TTFF
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CS-708 20
Example – 10b
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CS-708 21
Understanding Statements – 11c
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CS-708 22
Example – 11b
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CS-708 23
Compound Statement – 12a
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CS-708 24
Symbolic Representation – 13a
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CS-708 25
Logical Connectives – 14a
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CS-708 26
Examples – 14b
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CS-708 27
Translating from English to Symbols – 15
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CS-708 28
Translating from English to Symbols – 16a
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CS-708 29
Translating from English to Symbols – 16
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CS-708 30
Translating from English to Symbols – 17a
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CS-708 31
Translating from English to Symbols – 17b
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CS-708 32
Negation – 19
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CS-708 33
Truth Table for ~p – 20
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CS-708 34
Conjunction – 21
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CS-708 35
Truth Table for p ^ q – 22
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CS-708 36
Disjunction – 23
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CS-708 38
Truth Table
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CS-708 39
Truth Table for ~p^q - 2
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CS-708 40
Truth Table for ~p^q – 2a
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CS-708 41
Truth Table for ~p^q – 2b
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CS-708 42
Truth Table for ~p^q – 2c
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CS-708 43
~p ^ (q v~ r) – (2 - 3a)
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CS-708 44
~p ^ (q v~ r) – 2 - 3b
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CS-708 45
~p ^ (q v~ r) – 2 - 3c
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CS-708 46
~p ^ (q v~ r) – 2 - 3d
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CS-708 47
Truth Table for ~p (p v~ q) – 2 - 3e
v
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CS-708 48
Truth Table for (pvq) ^~ (p^q) – 2 - 4a
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CS-708 49
Truth Table for (pvq) ^~ (p^q) – 2 - 4c
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CS-708 50
Truth Table for (pvq) ^~ (p^q) – 2 - 4e
vv
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CS-708 51
Truth Table for (pvq) ^~ (p^q) – 2 -4f
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CS-708 52
Exclusive OR – 2 - 5
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CS-708 53
Symbols for Exclusive OR – 2 - 5a
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CS-708 54
Logical Equivalence – 2 - 6
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CS-708 55
Double Negation ~(~p) ≡ p – 2 - 7
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CS-708 56
Examples – 2 - 12
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CS-708 57
Example – 2 - 17c
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CS-708 58
Example – 2 - 17e
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CS-708 59
De Morgan’s Laws – 2 - 9
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CS-708 60
De Morgan’s Laws – 2 - 9a
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CS-708 61
Proof – 2 - 16
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CS-708 62
Proof – 2 - 16d
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CS-708 63
Application – 2 - 10
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CS-708 64
Exercise – 2 - 19
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CS-708 65
Tautology – 2 - 21
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CS-708 66
Example – 2 - 21a
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CS-708 67
Contradiction – 2 - 22
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CS-708 68
Example – 2 - 22a
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CS-708 69
Exercise – 2 - 23
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CS-708 70
Exercise – 2 - 24
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CS-708 71
Laws of Logic – 2 - 25
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CS-708 72
Laws of Logic – 2 - 25a
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CS-708 73
Laws of Logic – 2 - 25b
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CS-708 74
Laws of Logic – 2 - 25c
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CS-708 75
Laws of Logic – 2 - 25d
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CS-708 76
Application - 1
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CS-708 77
Example - 2
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CS-708 78
Simplifying a Statement – 3
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CS-708 79
Distributive Law in Reverse – 4
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CS-708 80
Exercise – 5
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CS-708 81
Exercise - 5a