002 - Module 1 Introduction to Theory of Computations

7/27/2019 002 - Module 1 Introduction to Theory of Computations

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INTRODUCTION

ITEU133

AUTOMATA AND THEORY OFCOMPUTATION


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SETS

- as a collection well defined objects havingcertain common property

- represented by a CAPITAL letter i.e. A, B, C,

METHOD OF SET NOTATION:

1. Roaster / Tabular Method

-the set is represented by actually listing the

elements which belong to it.- separated by comma (,) and enclosed between pair

of curly brackets { }

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SETS

Roaster / Tabular Method

A = { 1, 3, 5, 7}

Note:

1. The order of writing the elements of set isimmaterial

2. An element of a set is not written more than

once3. Roaster method is used only when the number

of elements in the set is finite

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SETS

2. Set Builder / Rule Methodsometimes is a set is defined by stating property

(P) which characteristics all the elements of the

set.

Example:

A= { x | x satisfies the property P }

A={x:P(x)}

A={x | x = 2n, n N}

Where P(x) means property P is satisfied byevery x of the set

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RECITATION

1. Apply Roaster method to represent the set of all

negative integers greater than -4

2. List the members of the following set:

A= { x | x is a real number such as x2 =1 }

3. Express in tabular form the set {b | b N, 3 b 8}

4. Express the set of all days in a week by set builder

method

5. Write the set of the letters in the word book

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TYPES OF SETS

1. Finite Set If A contains exactly n distinct

elements, where n is a non-negativeinteger, then A is a finite set and n is the

cardinalityof A

Ex:

A = { a, e, i, o, u}A = { x|x is an odd positive number less than 10}

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TYPES OF SETS

2. Infinite Set a set which is not finite is said to be

an infinite set

Ex:The set of natural number N = {1, 2, 3, 4,5,...}

3. Empty Set the set having no element / null setor void set. It is denoted by 0 or {}.

4. Unit Set a set containing only one element is

called a unit set or singleton set {0} , {a}

5. Subset If each element of the set A is also an

element of set B.

A = {1, 2, 3, 4} B={1, 2, 3, 4, 5, 6}

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TYPES OF SETS

6. Subset the set A is called the proper subset of B is

and only if each element of A is the element of B

and there is at least one element of B which is not

an element of the set A. (A B.)7. Family Set class of sets or the set of sets

ex. A = {0 , {a}, {a, b}}

8. Equal Set two sets A and B consisting of the same

elements

9. Equivalent set two sets are said to be equivalent,

it they contain the same number of elements

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TYPES OF SETS

10. Power Set set A is the set of all subset of set A.

it is denoted by P(A)

11. Disjoint Set the two sets A and B are disjoint, if

they have no element in common12. Universal Set

13. Cartesian product

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RECITATION

6. True/False. C={0} is an empty set

Categorize the following into finite or infinite set

7. A= {1, 1, 1, }

8. B= {x:x I and x is even}

Consider the following sets:

A = {a, b} B={a, b,c,d} C={a, e}

D={c, d, e} E={d, e}

Which of these sets are disjoint(2 points).

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SET OPERATIONS

Union The union of two sets is the set that has objects

that are elements of at least one of the two given

sets, and possibly both.

That is, the union of setsA and B, written A B, is aset that contains everything inA, or in B, or inboth.

Example: A = {1, 3, 9} B = {3, 5}

Therefore, AB = { ? }

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SET OPERATIONS

IntersectionThe intersection of setsA and B, written AB, is

a set that contains exactlythose elements thatare in bothA and B.

A B = {x : x A and xB}Example: A = {1, 3, 9}, B = {3, 5}, C = {a, b, c}

A B = { ? }

A C = { ? }

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SET OPERATIONS

Set DifferenceThe set difference of setA and set B, written as

AB, is the set that contains everything that is inA but not in B.

Given: A = {1,3,9}, B = {3,5} A-B B - A

Answer

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SET OPERATIONS

ComplementThe complement of setA, written as A is the setcontaining everything that is not inA.

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PROPERTIES OF SET OPERATIONS


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Example


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Given sets A and B are the subsets o f

a un iversal set U,prove that


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SEATWORK#1

Proof the following: De Morgans Law

1. ( A U B ) = A B

2. (A B) = A U B

Determine the set P, Q, AND R if the given that

P U Q = { 1, 2, 3, 4} P U R = {2, 3, 4, 5}

P Q = { 2, 3} P R = { 2, 4}


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VENN DIAGRAM

Out of forty students, 14 are taking English Compositionand 29 are taking Chemistry. If five students are in both

classes, how many students are in neither class? How

many are in either class? What is the probability that a

randomly-chosen student from this group is taking only

the Chemistry class?

Universal

Eng

Chem

514 - 5 = 9 29 - 5 = 24

40 - 9 - 5 - 24 = 2


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VENN DIAGRAM

There were 60 students in a class.25 students attend Woodshop class

and 20 students attended Word

Processing class.If 7 students were in both the classes,

how many students were not enrolled in

both Woodshop and Word Processingclass?


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SEATWORK (CLASS ACTIVITY)Survey of faculty and graduate students at the FEA-EAC

films school revealed the following information:

51 admire Moe 49 admire Larry

60 admire Curly 34 admire Moe and Larry

32 admire Larry and Curly 36 admire Moe and Curly

24 admire all three of the Stooges

1 admire none of the three stooges

a. How many people were surveyed?

b. How many admire Curly but not Larry nor Moe?

c. How many admire Larry or Curly?

d. How many admire exactly one of the Stooges?

e. How many admire exactly two of the Stooges?


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SEATWORK#3

A nutritionist at a school is planning a schedule ofbreakfasts for 175 students.

73 students say they want milk,

97 want Juice, and 60 want fruit.19 say they want both milk and fruit;

of these 11 want juice as well.

34 want only milk and 36 want only fruit.How many students want juice only?


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RELATIONS

A relation on sets S and Tis a set of orderedpairs (s, t), where

(a) s S (s is a member ofS )

(b) tT(c) S and Tneed not be different

(d) The set of all first elements in the

domain of the relation, and(e) The set of all second elements is the

range of the relation.


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RELATIONS

Then a relation on S and TisR= {(a, y), (c, w), (c, z), (d, y)}

Types of Relat ion s

1. Binary Relation2. Equality Relation ex. A={a, b, c}

3. Universal Relation

4. Void Relation

5. Inverse Relation ( x > y y < x)

ex. A = { 2, 3, 4, 5} B={2, 4, 6} A< B


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RELATIONS

Then a relation on S and TisR= {(a, y), (c, w), (c, z), (d, y)}

Equ ivalence Relat ion

A subset RofA A is called an equivalence relation onAifRsatisfies the following conditions:

(i) (a, a) Rfor all a A (Ris reflexive)

(ii) If (a, b) R, then (b, a) R, then (a, b) R(Rissymmetric) ex. A={2, 3, 4, 5} is a double of

(iii) If (a, b) Rand (b, c)R, then (a, c) R(Ristransitive)


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RECITATION

Determine if antisymmetric or symmetricA = {a, b, c}

1. R = { (a,b), (b, c), (a, c), (b, a), (c, a), (c, b)}

2. R = {(a,b), (b, c), (a, c) }


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REPRESENTATION OF RELATIONS

Relation as arrow diagram

Relation as table

Relation as a directed graph

Relation as a matrix

example for arrow diagram / table / matrix

A = {1, 2, 3, 4} B={a, b, c}R={(1, b), (1, c), (3, b), (4, a), (4, c)}


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REPRESENTATION OF RELATIONS

Example for directed graph

A = {1, 2, 3, 4}

R={(1, 1), (1, 2), (2, 1), (2, ,2), (3, 2), (3, 4),

(4, 3), (4, 4)}


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FUNCTIONS AND RELATIONS

Every function is a relation.

Not all relations are functions.

One-to-one correspondence function

One-to-many function

Many-to-one relation

Many-to-many relation For graphs of equations Vertical Line

Test


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GRAPHS AND TREES

A Graph is a bunch of vertices (or Nodes)which are represented by circles and are

connected by edges represented by lines.

Trees are undirected graphs. Any twovertices are connected by exactly one

simple path. A tree also doest not contain

a cycle.


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GRAPHS

A simple graph G = (V,E)

V = {set of all vertices} not empty

E = {set of all edges} not empty

E = {subsets of V with cardinality 2}

Digraph

In-degree

Out-degree

Isomorphism


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PROOF TECHNIQUES

The basic structure of a proof is easy: it isjust a series of statements, each one

being either

An assumption or

A conclusion, clearly following from an

assumption or previously proved result.


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PROOF TECHNIQUES

Direct Proof

Proof by Contradiction

Proof by Contrapositive

If and only if

Proof by Mathematical Induction


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DIRECT PROOF

Assume P then derive Q

Example:

Prove that Divisibility is Transitive

a, b and c are natural numbers

a is divisible by b and b is divisible by c

Prove that a is divisible by c

Try this: Prove that if a divides b and a dividesc then a divides b + c.


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PROOF BY CONTRADICTION

Assume not P then derive thecontradiction

Example:

Prove the Converse of the Pythagorean

Theorem


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ALPHABETS AND LANGUAGES

pattern matching in the command-lineshells

ls - in Unix

____ - in DOS

Regular expressions particular language for patterns


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ALPHABET


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STRINGS OVER AN ALPHABET


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STRINGS


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EXAMPLES

Documents

002 - Module 1 Introduction to Theory of Computations