CSC 361 Preliminaries 1

Preliminaries

Language: Set of Strings

CSC 361 Preliminaries 2

Specification and recognition of languages

Sets Operations Defining sets (inductively) Proving properties about sets (using induction) Defining functions on sets (using recursion) Proving properties about functions (using

induction)

Strings Operations

Languages : set of strings Operations

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Representation of Sets

Set: collection of objects Finite : can use enumeration

E.g., {a,b,c} Infinite : requires describing an

infinite characteristic of members using finite terms

E.g., { n N | even(n) /\ square(n) }

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Sets: Examples• Sets are denoted by { <collection of elements> }

•Examples:

{}{a,b}{{}}{1, 2, …, 100}

{0, 1, 2, …}

{0,2,4, …}{2n | n }

“the empty set”“the set consisting of the elements a and b”“the set consisting of the empty set”“the set consisting of the first 100 natural numbers”

“the set of all natural pair numbers”“the set of all natural pair numbers”

“the set consisting of all natural numbers”

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Operations on Sets

Set Inclusion and Set EqualityDefinition: •Given 2 sets, A and B, A is contained in B, denoted by A B, if every element in A is also an element in B

True or false:{e,i,t,c} {a, b, …, z}

for any set A, A A

for any set A, A {A}

true

true

false

Definition: •Given 2 sets A and B, A is equal to B, denoted by A = B, if A B and B A

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Review: Member Union Intersection Subset Powerset Cartesian product

Operations on Sets

BA x

},...}{,{

,

a

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Set Difference

DeMorgan’s Laws

Operations on Sets

X Y Y X

Y} z X { z | z Y X

YXYX

YXYX

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Operations on Sets

Cartesian Product• Definition: Given two sets, A and B, the Cartesian product of A and

B, denoted by A B, is the following set:

{(a,b) | a A and b B}

Examples:

•What is: {1, 2 , 3} {a,b} =•True or false: {(1,a), (3,b)} {1, 2 , 3} {a,b} •True or false: {1,2,3} {1, 2 , 3} {a,b}

truefalse

pair or 2-tuple

{(1,a), (1,b), (2,a), (2,b), (3,a), (3,b)}

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Operations on SetsCartesian Product of More Than 2 SetsDefinition: Given three sets, A, B and C, the Cartesian product of A, B, and C denoted by A B C, is the following set: {(a,b,c) | a A, b B, c C}

Definition. (x,y,z) = (x’,y’, z’) only if x = x’, y = y’ and z = z’

These definitions can be extended to define the Cartesian product: A1 A2 … An

and equality between n-tuples

Triple or 3-tuple

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Operations on Sets

• Some Cartesian product problems to try out:

• What is: {1, 2 , 3} {a,b} {,} =

• What is the form of the set A B C D

• What is the form of the set A B (C D)

• What is the form of the set (A B ) (C D)

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Partition of a Set X The set of subsets of X is a partitionpartition of X

iff(1) “covers all subsets”

(2) “pair-wise disjoint”

Collectively Exhaustive / Mutually Exclusive

},...,,{ 21 nXXX

XXXX n ...21

)()( ji XXji

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Examples Set of Natural numbers is

partitioned by “mod 5” relation into five “equivalence classes”:{ {0,5,10,…}, {1,6,11,…}, {2,7,12,…},

{3,8,13,…}, {4,9,14,…} } “String length” can be used to

partition the set of all bit strings.{ {},{0,1},{00,01,10,11},{000,

…,111},… }

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RelationsDefinition: Given two sets, A and B, A relation R is any subset of A B. In other words, R A B

• Relations are used to describe how elements of different sets interactQuestion: What does the relation A B indicate?

Examples of relations

Simple:•{(c,s) | c is a class at KU, s is a student at KU, AND s is in C }

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Relations

Example: Let I be the set of all classroom instructors at Kutztown Let S be the set of all students at Kutztown Then, we can define the relation Students of an Instructor,

over I x S, as follows:Students of an Instructor=

{(i,s) | i I, s S, s is in a course of instructor i}Questions:1. Does this relation represent an equivalence relation?2. Suppose this instructor teaches courses c1, c2, and c3.

Would the sets Students in instructor i’s course, for courses c1, c2, and c3, be disjoint?

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Equivalence RelationsA relation R is an equivalence relation if R is reflexive, symmetric, and transitive

Relation R is:• reflexive if (a,a) R for each a in the language• symmetric if when (a,b) R then (b,a) R • transitive if (a,b) R and (b,c) R, then (a,c) R

Equivalence relations are generalizations of the equality relation

{(1,2),(1,3),…, (2,3),(2,4),…}

{(a,b) | a and b are males between 35 and 40 years old}

“the relation x < y”x

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FunctionsDefinition: A function f from a set A to a set B, denoted by f: A B , is rule of correspondence between A and B such that there is a unique element in B assigned to each element in A i.e. for each a A there is one and only one b B such that (a,b) f

Note: •A is the domain of f•B is the range of f

{(c,s) | c is a class at KU, s is a student at KU, AND s is in C }

Question: Is the following relation also a function from the set of KU classes to the set of KU students?

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Functions

Kinds of Functions Total

All domain elements map to an element in the range

One to one (injection) Total, and each mapping from domain to

range is unique Onto (surjection)

All elements of the range are mapped to from a domain element

1-1 & Onto (bijection)

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(Total) Function

BAf :

A B

f

Domain Co-domain ( Range)

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One to One Function (injection)

BAf :

A B

f

Domain Co-domain ( Range)

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Onto Function (surjection)

BAf :

A B

f

Domain Co-domain ( Range)

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1-1 Onto Function (bijection)

BAf :

A B

f

Domain Co-domain ( Range)

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Inductive Definitions Constructive Example

Set of natural numbers N = {0,1,2,3,…}

Finite representation in terms of Seed element: zero Closure Operation: successor function {0,s(0),s(s(0)),s(s(s(0))),…}

Imposes additional structure on the domain.

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Basis case: Recursive step:

Closure: only if it can be obtained from 0 by a finite number of applications of the operation s.

(* Minimality condition to uniquely determine N *)

0

)(nsn

n

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Languages

A language can be specified as a set of strings.

Languages are classified by how they can be formed Regular languages are the most

constrained Others

Context-free Context-sensitive Recursively enumerable

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Language Semantics - Syntax

Regular sets/languages Generator

Regular Expressions, Regular Grammars Recognizer

Finite State Automata Context-free languages

Generator Context-free Grammar

Recognizer Push-down Automata