lecture 1.1: course overview, and propositional logic cs 250, discrete structures, fall 2013 nitesh...
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Lecture 1.1: Course Overview, and Propositional Logic
CS 250, Discrete Structures, Fall 2013
Nitesh Saxena
Adopted from previous lectures by Cinda Heeren
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Outline
Administrative Material Introductory Technical Material
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Some Important Pointers Instructor: Nitesh Saxena
Office: CH 133 Email: saxena@cis.uab.edu (best way to reach me!) Phone No: 205-975-3432 Office Hours: Thursdays 3-4pm (or by appointment)
Course Web Page (also accessible through my web-page)
http://www.cis.uab.edu/saxena/teaching/cs250-f13/
TA/Grader: Lutfor Rahman: lrahman@uab.edu, M.S. student Office location: Ugrad lab (CH 154) Office Hours: Mondays and Wednesdays 12:30-
1:30pm Blackboard:
https://cms.blazernet.uab.edu/cgi-bin/bb9login
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About the Instructor Associate Professor, CIS PhD graduate from UC Irvine Previously an Assistant Professor at the
Polytechnic Institute of New York University
Research in computer and network security, and applied cryptography
Web page: http://cis.uab.edu/saxena Research group (SPIES):
http://spies.cis.uab.edu
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Prerequisites
1. Fundamental course for anyone intending to become a computer scientist
2. MA 106 (pre-calculus trigonometry) OR3. MA 107 (pre-cal algebra/trigonometery)
OR4. MA 125 (calculus I) OR 5. MA 126 (calculus II) OR6. MA 227 (calculus III)7. Minimum grade of C
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What to expect The course would be quite involved and
technical Lot of mathematics Busy schedule
The grading will be curved I would love to give all A’s but I won’t mind giving F’s
when deserved I might/will make mistakes
Please point them out Talk to me if you have any complaints (or send me an
anonymous email ) But, I guarantee that
I will encourage you to do your best You’ll have fun I’ll help you learn as much as I can – don’t hesitate to
ask for help whenever needed
7
What I expect of you Please do attend lectures Take notes Review lecture slides after each lecture Solve text book exercises as you read through the
chapters Ask questions in the class Ask questions over email Attend office hours Try to start early on homework assignments
Don’t wait until the very last minute! Follow the instructions and submit assignments on
time
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Course Textbook Discrete Mathematics and its
Applications -- Kenneth Rosen Seventh Edition
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Grading 40% - 5 homework assignments
HW due every 15-20 days or so 60% - Exams
2 mid-terms: 30% (15% each) 1 final: 30%
Mid-Term 1 – Oct 1st or 2nd week (tentative) Mid-Term 2 – Nov 1st or 2nd week (tentative) Final – Dec 10
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Policies Against Cheating or Misconduct You are not allowed to collaborate with any other
student, in any form, while doing your homeworks, unless stated otherwise; perpetrators will at least fail the course or disciplinary action may be taken
No collaboration of any form is allowed on exams You can definitely refer to online materials and
other textbooks; but whenever you do, you should cite so in your homeworks. This is a rule of thumb.
Also check: https://www.uab.edu/students/academics/honor-code
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Late Homework Policy None – no late homeworks are
allowed Either you submit on time and your
homework will be graded OR you submit late and the homework is NOT graded
You should stick to deadlines, please Exception will be made ONLY under
genuine circumstances
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Tentative Course Schedule 1. Logic and Proofs (Chap 1) – ~5 lectures
1. Propositional Logic (1.1, 1.2)2. Equivalences (1.3)3. Quantifiers and Predicates (1.4, 1.5)4. Proof Techniques (1.7. 1.8)
2. Basic Structures (Chap 2) – ~5 lectures1. Sets and Set Operations (2.1, 2.2)2. Functions (2.3)3. Sequences and Summations (2.4)4. Matrices (2.6)
3. Induction and Recursion (Chap 5) – ~6 lectures1. Induction (5.1)2. Strong Induction and Well Ordering (5.2)3. Recursion and Structural Induction (5.3)4. Recursive Algorithms (5.4)5. Program Correctness (5.5)
4. Relations (Chap 9) – ~5 lectures1. Relations and Properties (9.1)2. Closures and Equivalence (9.4, 9.5)3. Partial Orderings (9.6)
5. Graphs (Chap 10) – ~5 lectures1. Graphs, Terminologies, and Models(10.1, 10.2)2. Isomorphism and Connectivity (10.3, 10.4)3. Paths and Shortest Path Problem (10.5, 10.6)4. Planar Graphs and Coloring (10.7, 10.8)
6. Miscellaneous, if time permits – ~4 lectures1. Counting (Chap 6)2. Trees (Chap 11)3. Number Theory (Chap 4)
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Scheduled Travel Usually conference and invited talks
travel Usually no class the week of travel However, this will not affect our overall
course schedule and topic coverage (perhaps a guest lecturer will cover on my behalf)
Information about any travel will be provided as it becomes available
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Instructions HW submissions
Name your files “Lastname_Firstname_HW#” Submit it on Blackboard
Please make sure that you have correctly submitted/uploaded the files (simply “saving” them may not be sufficient)
PDF format only Check the course web-site regularly
I am posting lecture slides and homeworks there Check your UAB email regularly
I am sending out announcements there e.g., when I post homeworks
Only use your UAB email to communicate with me and the TA
Please specify CS 250 to subject line so I can identify course
NO EXCUSES for not following instructions
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Proposition A proposition is a logical statement that is
either TRUE (T) or FALSE (F) Propositions (examples)
3+2=32 3+2=5 MA 106 is a prerequisite for CS 250 It is sunny today
Not Propositions (examples) What time it is now? X+1 = 2 Read this carefully What grade can I get in this course?
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Propositional Logic -- Negation
Suppose p is a proposition. The negation of p is written p and has meaning:
“It is not the case that p.” In English, it is referred to as a “NOT”
Ex. “CS173 is NOT Bryan’s favorite class” is a negation for “CS173 is Bryan’s favorite class”
Truth table for negation:p p
TF
FT
Notice that p is a
proposition!
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Propositional Logic -- Conjunction
Conjunction corresponds to English “AND”. p q is true exactly when p and q are both true.
Ex. “Amy is curious and clever” is a conjunction of “Amy is curious” and “Amy is clever”.
Truth table for conjunction:
p q p q
TTFF
TFTF
TFFF
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Propositional Logic -- Disjunction
Disjunction corresponds to English “OR” p q is true when p or q (or both) are true.
It is actually an “inclusive OR”
Ex. “Michael is brave OR nuts” is a disjunction of “Michael is brave” and “Michael is nuts”.
Truth table for disjunction:
p q p q
TTFF
TFTF
TTTF
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Propositional Logic – Exclusive OR
p q is true when only one of p or q is true
Ex: Students who have taken calculus or computer science, but not both, can enroll for this class.
Truth table for xor:
p q p q
TTFF
TFTF
FTTF
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Propositional Logic – Implication
Implication: p q corresponds to English “if p then q,” or “p implies q.”
If it is raining then it is cloudy If you have taken MA 106, you can enroll for CS 250 If you work hard then you can get a good grade
Truth table for implication:
p q p q
TTFF
TFTF
TFTT
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Propositional Logic – Biconditional
p q p q
TTFF
TFTF
TFFT
•This is equivalent to: (p q) (q p)• Also, referred to as the “iff” condition• For p q to be true, p and q must have the same truth value.
Truth table for biconditional:
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Complex Composite Propositionsand Equivalence Combination of many propositions using different
operations (negation, conjunction, disjunction, implication)
Precedence order for these operations:1. Negation2. Conjunction3. Disjunction4. Implies5. Biconditional
A complex proposition can often be reduced to a simple one
This means that the complex proposition and the simple proposition are logically equivalent
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Propositional Logic – Logical Equivalence
p is logically equivalent to q if their truth tables are the
same. We write p q. In other words, p is logically equivalent to q if p q is
True. We will study about equivalences more in the next
lecture But, for now, let us look at some examples
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Propositional Logic – Logical Equivalence
Challenge: Try to find a proposition that is equivalent to p q, but that uses only the connectives , , and .
p q p q
TTFF
TFTF
TFTT
p q p q p
TTFF
TFTF
FFTT
TFTT
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Distributive Law – an example of equivalence
Distributivity: p (q r) (p q) (p r)
p q r q r p (q r) p q p r (p q) (p r)
T T T T T T T T
T T F F T T T T
T F T F T T T T
T F F F T T T T
F T T T T T T T
F T F F F T F F
F F T F F F T F
F F F F F F F F
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Propositional Logic – special definitions
Contrapositives: p q and q p Ex. “If it is noon, then I am hungry.”
“If I am not hungry, then it is not noon.”Converses: p q and q p
Ex. “If it is noon, then I am hungry.” “If I am hungry, then it is noon.”
Inverses: p q and p q Ex. “If it is noon, then I am hungry.”
“If it is not noon, then I am not hungry.”
One of these things is not
like the others.
Hint: In one instance, the pair of propositions is
equivalent.
p q q p
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Propositional Logic – special definitions
A tautology is a proposition that’s always TRUE.
A contradiction is a proposition that’s always FALSE.
p p p p p p
T F
F T
T
T
F
F
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Propositional Logic – bit-wise operators All the operators are extensible and applicable to “bits”
and “bitstrings” ‘1’ is TRUE and ‘0’ is FALSE
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Propositional Logic – applications Computer Programs
Propositional logic is a key to writing good code…you can’t do any kind of conditional (if) statement without understanding the condition you’re testing.
Different programming languages may have different syntax for logic operators
Hardware and Gates: All the logical connectives we’ve discussed are also found in
hardware and are called “gates.” Foundational Element for Proof Systems and Proof Techniques
Ex: Classical proofs in provable cryptography based on counterpositives.
Logical searches Ex: “Alabama” and “Universities” Ex: “java – coffee”
Writing Policies Ex: firewall policies
Logical Puzzles and Games …
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Some Quick Questions- Is “what is your name” a proposition?- Is “The sun revolves around the earth” a proposition? If so,
what is its logical value.- What is the negation of “The sun revolves around the earth”?
What is the logical value of this negation.- “You can get a good grade if you perform well on homeworks
and you perform well on exams” – represent as a proposition.- “You may fail the course if you cheat or you do not attend any
lectures” – represent as a proposition.- What is p p equivalent to? - What is p p equivalent to?- What is:
- 1 1? - 0 1? - 1 0?
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Some Quick Questions If p is True and q is False, what is p -> q? Is p->q equivalent to p q? What will be the output of the following piece of
pseudocode:X = 50;If ( X > 35)
print “Pass”;Else
print “Fail”; How would (p q) (r s) be represented in computer
hardware (using gates)? What is the converse of: “if you are a good student, you
will end up getting a good grade”?
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