lecture 1.1: course overview, and propositional logic cs 250, discrete structures, fall 2013 nitesh...

Lecture 1.1: Course Overview, and Propositional Logic

CS 250, Discrete Structures, Fall 2013

Nitesh Saxena

Adopted from previous lectures by Cinda Heeren

04/19/23Lecture 1 - Course Overview,

and Propositional Logic 2

Outline

Administrative Material Introductory Technical Material



Some Important Pointers Instructor: Nitesh Saxena

Office: CH 133 Email: [email protected] (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: [email protected], 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



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



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



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



Course Textbook Discrete Mathematics and its

Applications -- Kenneth Rosen Seventh Edition



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



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



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



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)



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



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



Propositional Logic



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?



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!



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



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



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



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



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:



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



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



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



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



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



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



Propositional Logic – bit-wise operators All the operators are extensible and applicable to “bits”

and “bitstrings” ‘1’ is TRUE and ‘0’ is FALSE



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 …



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?



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”?



Today’s Reading and Next Lecture Rosen 1.1 and 1.2, and part of 1.3 Please start solving the exercises at the

end of each chapter section. They are fun.

Please read 1.3 and 1.4 in preparation for the next lecture

lecture 1.1: course overview, and propositional logic cs 250, discrete structures, fall 2013 nitesh...

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