1. Language: Set of Strings
Preliminaries
Language: Set of Strings
Examples: Names, dates, addresses, Java, C++, Patterns in PERL, Python, …
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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
Languages : set of strings
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Preliminaries

3. 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 e N | even(n) /\ square(n) }
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Preliminaries

4. Sets: Examples Sets are denoted by { <collection of elements> }{}
{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 consisting of all natural numbers”
“the set of all natural pair numbers”
“the set of all natural pair numbers”
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Preliminaries

5. Set Inclusion and Set Equality
Operations on Sets
Set Inclusion and Set Equality
Definition:
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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Preliminaries

6. Operations on Sets Review: Member Union Intersection Subset Powerset
Cartesian product
Definition of an op – its properties
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7. Operations on Sets Set Difference DeMorgan’s Laws CSC 361
Definition of an op – its properties
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Preliminaries

8. 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}
pair or 2-tuple
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}
{(1,a), (1,b), (2,a), (2,b), (3,a), (3,b)}
true
false
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9. Operations on Sets Cartesian Product of More Than 2 Sets
Definition: 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}
Triple or 3-tuple
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
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10. 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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11. Partition of a Set X The set of subsets of X is a partition of X iff
“covers all subsets”
“pair-wise disjoint”
Collectively Exhaustive / Mutually Exclusive
Combinatorics: list partition, number partition, …
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12. 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},… }
Equivalence = Similar for a purpose (approximation)
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13. Relations
Definition: 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 interact
Question: 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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14. 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:
Does this relation represent an equivalence relation?
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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15. Equivalence Relations
A 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 x
{(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” 
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16. Functions Definition: 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
Question: Is the following relation also a function from the set
of KU classes to the set of KU students?
{(c,s) | c is a class at KU, s is a student at KU, AND s is in C }
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17. Functions Kinds of Functions Total One to one (injection)
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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18. (Total) Function
Domain
Co-domain ( Range) A B f
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19. One to One Function (injection)
Domain
Co-domain ( Range) A B f
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20. Onto Function (surjection)
Domain
Co-domain ( Range) A B f
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21. 1-1 Onto Function (bijection)
Domain
Co-domain ( Range) A B f
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22. 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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23. (* Minimality condition to uniquely determine N *)
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 *)
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24. 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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25. Language Semantics - Syntax
Regular sets/languages
Generator
Regular Expressions, Regular Grammars
Recognizer
Finite State Automata
Context-free languages
Context-free Grammar
Push-down Automata
Identifiers, numbers, etc in a PL; Tokenizer
BNF; Parser
DTD; XML Parser
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