meljun cortes automata lecture review of important concepts 2

8/21/2019 MELJUN CORTES Automata Lecture Review of Important Concepts 2

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Theory of Computation (With Automata Theory)

* Property of STI

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Review of Important Concepts

Review of Important

Concepts

Review of Set Theory

Sets

• A set is a collection of objectssuch as numbers, letters,symbols, or other sets.

• These objects are called themembers or elements of theset.

• For example:

If set A = {0, 2, 4, 8, 16},then the members orelements of set A are 0, 2,4, 8, and 16.


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• In sets, the ordering andrepetition of the members arenot important.

Therefore, the set {1, 2, 3} isthe same as the set {2, 3, 1}.

Similarly, the set {1, 2, 3, 2} isthe same as the set {1, 2, 3}.

Set Membership

• The symbol is used todenote membership in a setwhile the symbol is used todenote non-membership in aset.

• For example:

If set A = {0, 2, 4, 8, 16},then 8 A while 7 A.


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Subsets

• For two sets A and B, B is asubset of A, if every member of B is also a member of A.

• Set B is a subset of set A is

denoted by B A.

• For example:

If set A = {0, 2, 4, 8, 16} andset B = {2, 8}, then B A.


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Infinite Set

• An in f in i te set is a set thathas infinitely many members.

• For example:

The set of integers

Z = {…, -2, -1, 0, 1, 2, …}

is an infinite set.

Empty Set

• An empty set is a set thatcontains no members.

• If set A is an empty set, it is

written as A = { } or A = .


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Union and Intersection of Sets

• The un ion of two sets A and Bis obtained by forming a singleset that contains all theelements in A and B.

The union of sets A and B iswritten as A B.

For example:

If

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

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


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• The in tersect ion of two sets A

and B is obtained by forming asingle set that contains all theelements that are in both Aand B.

The intersection of sets A andB is written as A B.

For example:

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

then A B = {3, 4}

If A B = (they have nocommon elements), then sets A and B are said to bedis jo in t .


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Complement of Sets

• Assume set A is a subset of set B ( A B). Thecomplement of A with respectto B is the set of all elements

in B that are not in A.

The complement of set A is

written as Ā or A .

For example:

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

then A = {1, 2, 5, 6}


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Set-Builder Notation

• The set-builder notation isused to describe a setcontaining elements that has a

certain property or follow acertain rule.

The following designations areused in conjunction with theset-builder notation:

N = set of natural numbers

Z = set of all integers

R = set of all real numbers


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For example:

A = { x x Z , 1 0


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The set-builder notation is

often used to describe infinitesets.

Examples:

A = { x x N , x ≥ 0}

This is read as“Set A is the set of all x such that x is a naturalnumber and is greater thanor equal to 0.”

A = { x x = y 2, y N }

This is read as“Set A is the set of all x such that x is equal to y 2

where y is a natural

number. Set A is thereforethe set of perfect squares.”


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Power Sets

• The power set of A is the setof all subsets of set A.

• The power set of A is writtenas P ( A) o r 2 A.

• For example:

Let set A = {1, 2, 3}.

P ( A) = { , {1}, {2}, {3},{1,2}, {2, 3}, {1, 3},{1, 2, 3}}


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Sequences and Tuples

• Sequences are similar to setsexcept for the fact that theordering and repetition of the

members in sequences areimportant.

• For example:

The sequence {1, 2, 3} isdifferent from the sequence{2, 3, 1}.

The sequence {1, 2, 3, 2} isdifferent from the sequence{1, 2, 3}.


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• Sequences may be finite or infinite.

• Finite sequences are calledtuples .

• A sequence with k elements is

a called a k -tuple.

For example:

The sequence {1, 2, 3} is a3-tuple while the sequence{a, b} is a 2-tuple.

• A 2-tuple is often called a pair .


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Cartesian Product of Sets

• The Cartes ian Pro duc t , or Cro ss Produ ct of two sets Aand B, is the set of all 2-tuplesor pairs wherein the first

element of a pair is a member of set A and the secondelement of a pair is a member of set B.

• The Cartesian product of sets A and B is written as A B.

• Examples:

Let set A = {1, 2}and set B = {x, y}

A B = { {1, x}, {1, y},{2, x}, {2, y} }

B A = { {x, 1}, {x, 2},

{y, 1}, {y, 2} }


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Letset A = {1, 2, 3}set B = {x, y}

A B = { {1, x}, {1, y},{2, x}, {2, y},{3, x}, {3, y} }

B A = { {x, 1}, {x, 2}, {x, 3},{y, 1}, {y, 2}, {y, 3} }

A A = { {1, 1}, {1, 2}, {1, 3},{2, 1}, {2, 2}, {2, 3},{3, 1}, {3, 2}, {3, 3} }

B B = { {x, x}, {x, y},{y, x}, {y, y} }


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• Generally, the Cartesian

product of k sets A1, A2 , …, Ak ,is the set of all k -tuples (a1, a2 ,…, ak ) where ai Ai .

• For example:

Letset A = {1, 2, 3},set B = {x, y}, andset C = {$, %}

A B C = {{1, x, $}, {1, x, %},{1, y, $}, {1, y, %}, {2, x, $},{2, x, %}, {2, y, $} {2, y, %},{3, x, $}, {3, x, %}, {3, y, $},{3, y, %}}

C B C = {{$, x, $},{$ , x, %}, {$, y, $},{$, y, %}, {%, x, $},{% , x, %}, {%, y, $},

{%, y, %}}


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Functions and Relations

Functions

• A func t ion is an object thatdefines the relation between aknown input and the output it

produces.

• Functions are usually writtenas f ( x ) = y . For a function f , if the input is x then the output isy .

• Examples:

Given a functionf ( x ) = x + 1,

if the input x = 2, then theoutput is f ( x ) = 3.

Given a function f ( x ) = x2,if the input x = 2, then theoutput is f ( x ) = 4.


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• For any given function, thesame input always producesthe same output.

• The set of all possible valuesthe input of a function can take

on is called the domain of thefunction.

• The set of all possible outputvalues of a function is calledthe range of the function.

• For example:

For the function f ( x ) = x + 1,the domain and range are

the set of all real numbers.

For the function f ( x ) = x 2,the domain is the set of allreal numbers while therange is the set of non-

negative real numbers.


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• A function is also called amapping .

For example:

For the function f ( x ) = x + 1,

if x = 2, then f ( x ) = 3.

It can then be said that thefunction f ( x ) maps 2 to 3.

The function also maps3 to 4, 4 to 5, etc.

• Therefore, a function is somerule that associates to eachelement in its domain some

element in its range.

For example, the functionf ( x ) = x 2 maps each realnumber to its square.


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Binary Relations

• A b inary relat ion from set Ato set B is the set of orderedpairs { x , y } where x A andy B.

• The ordered pair { x , y } is in arelation R if element x isrelated to element y as definedby R .

• For example:

Let A = {3, 4, 5, 6, 7, 8, 9}and B = {1, 2, 3}.

If R is defined as x = y 2, the

relation R is:

R = {{4, 2}, {9, 3}}

• If the ordered pair { x , y } R ,then we write a xRy .


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• A binary relation may involveonly one set, A. If that is thecase, then the relation issimply a relation on set A.

• For example:

Let set A = {1, 2, 3}.

If R is defined as x < y , therelation R on set A is:

R = {{1, 2}, {1, 3}, {2, 3}}

Take note that the other

ordered pairs {1, 1}, {2, 1},{2, 2}, {3, 1}, {3, 2}, and {3,3} are not included in R because they do not satisfy x < y .


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Properties of Binary Relations

• A relation R on set A isref lexive if for every x A, xRx . In other words, eachelement of set A is related to

itself.

For example:

The relation “is equal to” isreflexive.

However, the relation“is less than” is not reflexivesince 1 < 1 is not possible.The relation cannot be

applied to the sameelement.

The relation “looks thesame as” is reflexive sinceany person looks like

himself.


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• A relation R on set A issymmetr ic if for every x, y A, xRy implies yRx .

For example:

The relation “is equal to” issymmetric.

However, the relation“is less than” is notsymmetric since 1 < 2 doesnot imply 2 < 1.

The relation “is a sibling of ”is symmetric since if John isa sibling of Peter, then itimplies that Peter is asibling of John.


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• A relation R on set A ist ransi t ive if for every x, y , z A, xRy and yRz implies xRz .

For example:

The relation “is less than” istransitive since if 1 < 2 and2 < 3, then it implies that 1< 3.

However, the relation “is thesquare of ” is not transitivesince 16 is the square of 4and 4 is the square of 2does not imply that 16 is thesquare of 2.

The relation “is taller than”is transitive since if John istaller than Peter and Peteris taller than Paul, then itimplies that John is taller

than Paul.


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• A binary relation R is anequ ivalence relat ion if R isreflexive, symmetric, andtransitive.

For example:

Is the relation “looks thesame as” an equivalencerelation?

Check if the relation isreflexive, symmetric, andtransitive.


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It is reflexive since anyperson looks the same ashimself.

It is symmetric since if John

looks the same as Peterthen it implies that Peterlooks the same as John.

It is transitive since if Johnlooks the same as Peterand Peter looks the sameas Paul, then it implies thatJohn looks the same as

Paul.

The relation “looks the

same as” is therefore an

equivalence relation.

meljun cortes automata lecture review of important concepts 2

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