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CSCE 551:

Chin- ser Huanghuangct@cse.sc.edu

University of South Carolina

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A M

Chin- ser Huan

Ph.D. in Computer Sciences, University ofTexas at Austin

Research in network security, network protocol

design and verification, distributed systems My web page can be found at

http://www.cse.sc.edu/~huangct

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A h r A graduate/undergraduate course focusing on

answer ng e o ow ng ques on:

What are the fundamental capabilities and

We will study three areas Automata: a formal mathematical model of a

computational device that can do rudimentary patternmatching in a string

Complexity: Why are problems easy? Hard?

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A h r We will cover the fundamental knowled e

you SHOULD know

to learn

,interaction between you and me

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r Inf rm i n nlin htt ://www.cse.sc.edu/~huan ct/CSCE551

S10/index.htm

Links to handouts and other useful links are

Lecture slides will be available online too

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Y r B r Come to every lecture and take notes in class

Keep yourself exposed to interestingcomputational problems, theorems, and their

Finish each assigned reading and participate thediscussion

Do not wait till last minute to work onassignments or prepare for exam

Enjoy the fun!

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Asetis a rou of ob ects called elements or members

of this set. For example, the students in this room form aset.

se can e e ne y s ng a s e emen s ns ebraces, e.g.: S= {7, 21, 57}

For lar e or infinite sets an alternative wa of definin aset is by giving some criterion for membership in the set,e.g.: S= {n| nis an even integer}

e or er an repe ons o e emen s n se s o nomatter for example,

= =

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he membershi is denoted b s mbol. For exam le

21 Sbut 10 S. Two sets are equal if they have exactly the same

mem ers.

For two setsAand B, we sayAis a subsetofBand writeA Bif ever member ofAis also a member ofB.

We say thatAis aproper subsetofBand writeA BifAis a subset ofBand not equal to B.

he set of all subsets of a setAis called thepower setofAand denoted 2A.

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many of them are proper?

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Ex m l f The set with no elements is called the empty set

and denoted . The empty set is a subset of anyother set.

e set o natura num ers or N :

N= {1, 2, 3, ...} he set of integers Z (or Z):

Z= {, -2, -1, 0, 1, 2, }

It is clear that N Z.

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Ex m l f The set of perfect squares {n| n= m2 for some

m N

} is a subset of bothN

andZ

. The number of elements in a setAis called the

cardinality(or size) of the set and denoted by |A|.

We have || = 0 and |N| = |Z| = . Q: For a finite setA, what is |2A|?

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r i n For given two setsAand B, one can define the

following set operations: Union:A B. Example: {1, 2, 3} {1, 3, 5} = {1, 2, 3,

.

Intersection:A B. Example: {1, 2, 3} {1, 3, 5} =

1 3 .

Difference:A\B. Example: {1, 2, 3} \ {1, 3, 5} = {2}.

In the case ofB A, the result ofA\Bis also

called the complement ofBinA.

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V nn Di r m he Venn dia ram is a convenient wa to

illustrate the set operations.

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Ase uenceis a list of ob ects in some order. For

example, sequences of the students names in alphabeticorder such as (Alice, Bob).

n con ras o se s, repe ons an or er ma er nsequences. For example, the sequences (7, 21, 57) and

(7, 7, 57, 21) are not equal.

Finite sequences are called tuples. In particular, asequence with kelements is called k-tuple (as well as

, , , .

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All k-tu les x x x where x is taken from the set

Ai , form a setA1 A2 Ak , called the Cartesianproductor cross productof the setsA1, A2, , Ak .

or examp e, 1 = p, q an 2 = , , en

A1 A2 = {(p, 1), (p, 2), (p, 3), (q, 1), (q, 2), (q, 3)}

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Afunction ma in fsets u a corres ondence

between the elements of one setAand elements of theother set B, written f:A B.

n par cu ar, an e emen a correspon s o anelement b Bunder the function f, we write f(a) = b,

where ais called the input(or argument) and bis calledthe output(or value) off.

The setAis called the domain offwhile the set Bis

.

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For a finite setA the function f:A Bcan be defined

by a table that lists all possible inputs and gives theoutput for each input.

r x , r = 5 = , , , ,integers modulo 5), a function f: Z5 Z5 that adds 1 to

its input and then outputs the result modulo 5 can bespecified as

a 0 1 2 3 4

a

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A function f:A A A Bis called a k-ar function.

Unary functionscorrespond to the case k= 1 while binaryfunctionscorrespond to the case k= 2.

,property. For example, the property evendefines evenness of

a given integer: even(4) = TRUE and even(5) = FALSE. proper y : , , w ose

domain is the set ofk-tuples, is called k-ary relation(onA).An example is the beatsrelation between scissors, paper, and

s one see p. n pser . For binary relation R, we often use infixnotation writing xRy

instead ofR(x, y) = TRUE.

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A binar relation Ron a setAis called an e uivalence

relation ifRis: Reflexive: for every x A, xRx;

ymme r c: or every x, y , x y mp es y x;

Transitive: for every x, y, z A, xRyand yRzimplies xRz.

relation.

Q: What other equivalence relations you know?

Q: Is the relation less or equal () an equivalencerelation?

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Ann n m n Readin assi nment: Ch. 0

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