CS466 (Prasad) L1Sets 1

Introduction

Language: Set of Strings

CS466 (Prasad) L1Sets 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

CS466 (Prasad) L1Sets 3

Representation of Sets

• Set: collection of objects– Finite : can use enumeration

• E.g., {a,b,c}

– Infinite : requires describing characteristic property of members finitely

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

• COURSE GOAL: Study techniques for finitely describing specific families of sets

CS466 (Prasad) L1Sets 4

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

CS466 (Prasad) L1Sets 5

• Member, Union, Intersection, Subset, Powerset, Cartesian product, …

• Set Difference

• DeMorgan’s Laws

Operations on Sets

X Y Y X

Y} z X { z | z Y X

,...,,,,

YXYX

YXYX

CS466 (Prasad) L1Sets 6

Partition of a Set X

The set

of non-empty subsets of X is a partitionpartition of X iff(1) “covering”

(2) “pair-wise disjoint”

Collectively Exhaustive / Mutually Exclusive

},...,,{ 21 nXXX

XXXX n ...21

)()( ji XXji

CS466 (Prasad) L1Sets 7

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},… }

CS466 (Prasad) L1Sets 8

(Total) Function

BAf :

A B

f

Domain Co-domain ( Range)

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One-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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One to one correspondence 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.

CS466 (Prasad) L1Sets 13

• 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

CS466 (Prasad) L1Sets 14

URLs for Visualizing Recursion

• http://math.rice.edu/~lanius/fractals/• http://www-mickunas.cs.uiuc.edu/java-book.old/

code/ch11/applets/HilbertApplet.html• http://www-mickunas.cs.uiuc.edu/java-book.old/

code/ch11/applets/SierpinskiApplet.html• http://www-mickunas.cs.uiuc.edu/java-book.old/

solutions/ch11/SierpinskiGasket.html

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Recursive Definitions of Functions

• Addition– Basis case:– Recursive step:– Closure: …

• Multiplication– Basis case:– Recursive step:– Closure: …

mmm 0 :

)()( :, nmsnsmnm

00 : mm

nmmnsmnm *)(* :,

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Defining a Subset of Grid Points

y

x

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• Explicit Definition

• Implicit Definition

• Recursive Definition

),...}3,2(),2,1(),1,0(

),...,2,2(),1,1(),0,0{(

)1()( yxyx

0

1

0

}),(|)1,1{(

)}1,0(),0,0{(

ii

ii

LL

LjijiL

L

CS466 (Prasad) L1Sets 18

Other Recursive Definitions

0

1

0

}),(|)1,1{(

)}1,1(),1,0(),0,0{(

ii

ii

ML

MjijiM

M

}0,0{

}),(|)1,1{(

)}1,1(),1,0{(

0

1

0

ii

ii

NL

NjijiN

N

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Observations

• Recursive definition of a set typically contains a finite number of seed elements and a finite number of rules to generate successive sets of points, whose infinite union yields the set.

• Even though the various definitions “look” different, they capture the same set.

Equivalence Problem.

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Another Example

0

1

0

}),(|

)1,1(),,1{(

)}0,0{(

ii

i

i

LL

Lji

jijiL

L

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“Generation”

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Principle of Mathematical Induction

• Basis: PP holds for every element in

• Induction Step: If, whenever PP holds for every element in

PP also holds for every element in

then, by PMI,

PP holds for every element in

0L

,,...,, 10 iLLL1iL

L

0i

iLL

CS466 (Prasad) L1Sets 23

• Let be the Fibonacci number. Then, prove that

• Basis:

• Hypothesis:

• Prove:

nF thn

Example

110*1

)1( 12102

FFF

nnnn FFFn )1( :1 2

11

nFFFn )1( : 211

)1( :1 2

11 FFFn

CS466 (Prasad) L1Sets 24

• Proof

(Induction Hypothesis)

)1(

)1(

)1()(

)1()(

)1(

211

121

211

21

121

2111

1211

1212

FFF

FFFFF

FFFFF

FFFF

FFF n

CS466 (Prasad) L1Sets 25

Example• Strictly Binary Tree

• Single node.

• Every node is a leaf or has precisely two children.

• Prove that for all SBTs:2*leaves(T)-2 = arcs(T)

T1 T2

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(Induction on the number of nodes or height.)

Basis: 2*1-2=0

Induction Hypothesis:

For trees of height h < , the result holds.

Induction Step: Show it holds for trees of height

2*leaves(T1)-2=arcs(T1) (induction hypothesis)

2*leaves(T2)-2=arcs(T2)

2*(leaves(T1)+leaves(T2))-2-2=arcs(T1)+arcs(T2)

2*leaves(T)-2=arcs(T)

0h

0h