january 29, 2015applied discrete mathematics week 1: logic and sets 1 welcome to cs/math 320l –...

January 29, 2015 Applied Discrete Mathematics Week 1: Logic and Sets

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Welcome to

CS/MATH 320L – Applied Discrete Mathematics

Spring 2015

Instructor: Marc Pomplun


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Instructor – Marc PomplunOffice: S-3-171

Lab: S-3-135

Office Hours: Tuesdays 4:00pm – 5:30pm Thursdays 7:00pm – 8:30pm

Phone: 287-6443 (office) 287-6485 (lab)

E-Mail: [email protected]

Website: http://www.cs.umb.edu/~marc/cs320/


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The Visual Attention Lab

Eye movement research


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The new EyeLink-2K System


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Example: Distribution of Visual Attention


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Selectivity in Complex Scenes


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Modeling of Brain Functions


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Modeling of Brain Functionsunit and connectionin the interpretive network

unit and connectionin the gating network

unit and connectionin the top-down bias network

layer l +1

layer l -1

layer l


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Computer Vision:


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Human-Computer Interfaces:


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Now back to CS 320L:Course Kit: Kenneth H. Rosen, Discrete Mathematics and its Applications7th Edition(Available at the UMB Bookstore)

On the Web:http://www.cs.umb.edu/~marc/cs320/(contains all kinds of course information and also my slides in PDF and PPT formats, updated after each session)


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Your Evaluation

• 4 sets of exercises each set 5% 20%

(only individual submissions allowed)

• midterm (75 minutes) 35%

• final exam (2.5 hours) 45%


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Grading

95%: A 90%: A-

74%: C+ 70%: C 66%: C-

86%: B+ 82%: B 78%: B-

62%: D+ 56%: D 50%: D-

50%: F

For the assignments, exams and your course grade, the following scheme will be used to convert percentages into letter grades:


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Academic DishonestyYou are allowed to discuss problems regarding your homework with other students in the class.

However, you have to do the actual work (computing values, writing algorithms, drawing graphs, etc.) by yourself.

You cannot copy anything from other sources (Wikipedia, other students’ work, etc.)

The first violation will result in zero points for the entire homework or exam (and official notification).

The second violation will result in failing the course.


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Complaints about Grading

If you think that the grading of your assignment or exam was unfair,

• write down your complaint (handwriting is OK),• attach it to the assignment or exam, • and give it to me or put it in my mailbox.

I will re-grade the whole exam/assignment and return it to you in class.


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Why Care about Discrete Math?

• Digital computers are based on discrete “atoms” (bits).

• Therefore, both a computer’s – structure (circuits) and– operations (execution of algorithms)

can be described by discrete math.


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Syllabus • Logic and Set Theory• Functions and Sequences• Algorithms• Applications of Number Theory• Mathematical Reasoning• Counting• Probability Theory• Relations and Equivalence Relations• Graphs and Trees• Boolean Algebra


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Mathematical Appetizers

Useful tools for discrete mathematics:

• Logic• Set Theory• Functions• Sequences


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Logic• Crucial for mathematical reasoning• Used for designing electronic circuitry

• Logic is a system based on propositions.• A proposition is a statement that is either true or

false (not both).• We say that the truth value of a proposition is

either true (T) or false (F).

• Corresponds to 1 and 0 in digital circuits


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The Statement/Proposition Game

“Elephants are bigger than mice.”

Is this a statement? yes

Is this a proposition? yes

What is the truth value

of the proposition? true


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“520 < 111”




of the proposition? false


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“y > 5”


Is this a proposition? no

Its truth value depends on the value of y, but this value is not specified.

We call this type of statement a propositional function or open sentence.


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“Today is January 29 and 99 < 5.”




of the proposition? false


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“Please do not fall asleep.”

Is this a statement? no

Is this a proposition? no

Only statements can be

propositions.

It’s a request.


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“If elephants were red,

they could hide in cherry trees.”




of the proposition? probably false


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“x < y if and only if y > x.”




of the proposition? true

… because its truth value does not depend on specific values of x and y.


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Combining Propositions

As we have seen in the previous examples, one or more propositions can be combined to form a single compound proposition.

We formalize this by denoting propositions with letters such as p, q, r, s, and introducing several logical operators.


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Logical Operators (Connectives)

We will examine the following logical operators:

• Negation (NOT)• Conjunction (AND)• Disjunction (OR)• Exclusive or (XOR)• Implication (if – then)• Biconditional (if and only if)

Truth tables can be used to show how these operators can combine propositions to compound propositions.


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Negation (NOT)

Unary Operator, Symbol:

P P

true false

false true


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Conjunction (AND)

Binary Operator, Symbol:

P Q PQ

true true true

true false false

false true false

false false false


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Disjunction (OR)


P Q PQ

true true true

true false true

false true true

false false false


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Exclusive Or (XOR)


P Q PQ

true true false

true false true

false true true

false false false


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Implication (if - then)


P Q PQ

true true true

true false false

false true true

false false true


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Biconditional (if and only if)


P Q PQ

true true true

true false false

false true false

false false true


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Statements and OperatorsStatements and operators can be combined in any

way to form new statements.

P Q P Q (P)(Q)

true true false false false

true false false true true

false true true false true

false false true true true


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Statements and OperationsStatements and operators can be combined in any way

to form new statements.

P Q PQ (PQ) (P)(Q)

true true true false false

true false false true true

false true false true true

false false false true true


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Equivalent Statements

P Q (PQ) (P)(Q) (PQ)(P)(Q)

true true false false true

true false true true true

false true true true true

false false true true true

The statements (PQ) and (P)(Q) are logically equivalent,

because (PQ)(P)(Q) is always true.


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Tautologies and Contradictions

A tautology is a statement that is always true.

Examples: • R(R)• (PQ)(P)(Q)

If ST is a tautology, we write ST.

If ST is a tautology, we write ST.


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Tautologies and Contradictions

A contradiction is a statement that is always

false.

Examples: • R(R)• ((PQ)(P)(Q))

The negation of any tautology is a contra-

diction, and the negation of any contradiction is

a tautology.


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ExercisesWe already know the following tautology:

(PQ) (P)(Q)

Nice home exercise:

Show that (PQ) (P)(Q).

These two tautologies are known as De Morgan’s laws.

january 29, 2015applied discrete mathematics week 1: logic and sets 1 welcome to cs/math 320l –...

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