Lecture 1Overview

Lecture 1Overview

TopicsTopics1. Proof techniques: induction, contradiction Proof techniques

June 1, 2015

CSCE 355 Foundations of Computation

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Models of ComputationModels of Computation

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Summer ScheduleSummer Schedule

Tests on MondaysTests on Mondays June 8 June 15 June 22 1 hour long

Exam June 26Exam June 26

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Course OutcomesCourse Outcomes

Mathematical prerequisites: functions, relations, Mathematical prerequisites: functions, relations, properties of relations, posets. properties of relations, posets.

Proof TechniquesProof Techniques

Finite automata: regular languages, regular Finite automata: regular languages, regular expressions, DFAs, NFAs, equivalences. expressions, DFAs, NFAs, equivalences.

Limitations: pumping lemma Limitations: pumping lemma

Context free languages: grammars, push-down Context free languages: grammars, push-down automata automata

Turing machines: undecidability, the halting problem Turing machines: undecidability, the halting problem

Intractability: NP, NP-Completeness Intractability: NP, NP-Completeness

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PrerequisitesPrerequisites

CSCE 211CSCE 211 Number systems, Boolean algebra, logic design, sequential machines

Mealy machinesMoore machines

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PrerequisitesPrerequisites

CSCE 350CSCE 350 Techniques for

representing and processing information, including the use of

lists, trees, and graphs;

analysis of algorithms;

sorting, searching, and hashing techniques.

MATH 374MATH 374 Propositional and

predicate logic; proof techniques; recursion and

recurrence relations; sets, combinatorics, and probability; functions, relations, and matrices; algebraic structures.

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Review of Relations on SetsReview of Relations on Sets

Binary relations - (X, Y) Binary relations - (X, Y) ἐἐ R R or X Rel Y or X Rel Y < on integers likes (X,Y)

Unary relation - propertiesUnary relation - properties boring(matthews)

Ternary relationTernary relation “X was introduced to Y by Z” -- ( X, Y, Z) Table in a relational database

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Special types of RelationsSpecial types of Relations

InjectionsInjections

SurjectionsSurjections

FunctionsFunctions

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Properties of RelationsProperties of Relations

Property Def Example Neg-Example

Reflexive

Irreflexive

symmetric

antisymmetric

asymmetric

transitive

Total

Injection

Surjection

function

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PosetsPosets Partially Ordered Sets (POSETS)Partially Ordered Sets (POSETS)

Reflexive Antisymmetric Transitive

Hasse Diagram Hasse Diagram

Topological sorting Topological sorting

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Equivalence relationsEquivalence relations

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Proof TechniquesProof Techniques

1.1 Direct proof 1.2 Proof by induction 1.3 Proof by transposition 1.4 Proof by contradiction 1.5 Proof by construction 1.6 Proof by exhaustion 1.7 Probabilistic proof 1.8 Combinatorial proof 1.9 Nonconstructive proof 1.10 Proof nor disproof 1.11 Elementary proof

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Deductive ProofsDeductive Proofs

the conclusion is established by logically combining the conclusion is established by logically combining the axioms, definitions, and earlier theoremsthe axioms, definitions, and earlier theorems

Example: The sum of two even integers is even.Example: The sum of two even integers is even.

HypothesisHypothesis

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Theorem 1.3 used to prove Theorem 1.4Theorem 1.3 used to prove Theorem 1.4

Theorem 1.3 If x >= 4 then 2Theorem 1.3 If x >= 4 then 2xx >= x >= x22..

Theorem 1.4 If x is the sum of the squares of 4 Theorem 1.4 If x is the sum of the squares of 4 positive integers then 2positive integers then 2xx >= x >= x22..

ProofProof

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Theorem 1.3 If x >= 4 then 2x >= x2.Theorem 1.3 If x >= 4 then 2x >= x2.

f(x) = xf(x) = x22 / 2 / 2xx..

Then what is the derivative f’ of fThen what is the derivative f’ of f

Derivative of quotient??Derivative of quotient?? http://www.math.hmc.edu/calculus/tutorials/quotient_rule/

So f’(x) =So f’(x) =

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Proofs about Equality of SetsProofs about Equality of Sets

To prove S = TTo prove S = T Show S is a subset of T, and T is a subset of S

Commutative law of unionCommutative law of union

Theorem 1.10 Distributive law of union over Theorem 1.10 Distributive law of union over intersectionintersection

ProofProof

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Proof by ContradictionProof by Contradiction

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If and only If statementsIf and only If statements

IF H then C H = Hypothesis C = conclusionIF H then C H = Hypothesis C = conclusion H implies C H only if C C if H

A if and only if BA if and only if B If part : Only-if part

Theorm 1.7 ceiling = floor Theorm 1.7 ceiling = floor x is an integer x is an integer

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InductionInduction

Given a statement S(n) about an integer n that we Given a statement S(n) about an integer n that we want to prove.want to prove.

Basis Step: Show S(i) is true for a particular integer iBasis Step: Show S(i) is true for a particular integer i Usually i = 0 or i = 1

Inductive Step: Inductive Step: AssumeAssume S(n) is true S(n) is true for n >= i and for n >= i and then show S(n+1) is truethen show S(n+1) is true

Inductive Hypothesis:Inductive Hypothesis: Assume S(n) is true Assume S(n) is true

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Example Induction Proof: Theorem 1.16Example Induction Proof: Theorem 1.16

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Number of leaves in complete tree of height h is 2h. Number of leaves in complete tree of height h is 2h.

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More general inductionMore general induction

Basis step as beforeBasis step as before

Assume S(k) for all k <= n then show S(n)Assume S(k) for all k <= n then show S(n)

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Recursive Def of TreeRecursive Def of Tree

Basis: a single node is a tree.Basis: a single node is a tree.

If T1, T2, … Tk are trees then a new tree can be If T1, T2, … Tk are trees then a new tree can be formed byformed by1. Add new node N, the root of the new tree

2. Add copies of T1… Tk

3. Add an edge from N to the root of each T1, T2, … Tk

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Structural InductionStructural Induction

For objects with recursive definitions consisting of For objects with recursive definitions consisting of base objects and then combining rulesbase objects and then combining rules

Basis step: show the proposition S(X) holds for Basis step: show the proposition S(X) holds for every base object X.every base object X.

Inductive step: Given a recursive structure X formed Inductive step: Given a recursive structure X formed from X1, X2, … Xn by the application of the def. thenfrom X1, X2, … Xn by the application of the def. then

Assume S(X1) S(X2) …. S(Xn) are true and Assume S(X1) S(X2) …. S(Xn) are true and

show that S(X) is trueshow that S(X) is true

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Recursive Def of Arithmetic ExpressionsRecursive Def of Arithmetic Expressions

Basis: a number or a variable is an expression.Basis: a number or a variable is an expression.

If E and F are expressions then a new expression G If E and F are expressions then a new expression G can be formed by applying one of the three rulescan be formed by applying one of the three rules

1.1. G = E + FG = E + F

2.2. G = E * FG = E * F

3.3. G = ( E )G = ( E )

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Every Expression has equal number of left and right parenthsesEvery Expression has equal number of left and right parenthses

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HomeworkHomework

1.1. ..

2.2. Prove if a complete binary tree has n leaves then Prove if a complete binary tree has n leaves then it has 2n-1 nodes.it has 2n-1 nodes.

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References– Mathematical FoundationsReferences– Mathematical Foundations http://en.wikipedia.org/wiki/Binary_relationhttp://en.wikipedia.org/wiki/Binary_relation

http://en.wikipedia.org/wiki/Relation_(mathematics)http://en.wikipedia.org/wiki/Relation_(mathematics)

http://en.wikipedia.org/wiki/Mathematical_proofhttp://en.wikipedia.org/wiki/Mathematical_proof

http://en.wikipedia.org/wiki/Proofs_from_THE_BOOKhttp://en.wikipedia.org/wiki/Proofs_from_THE_BOOK

Extended “Proof” techniquesExtended “Proof” techniques

http://www.maths.uwa.edu.au/~berwin/humour/http://www.maths.uwa.edu.au/~berwin/humour/invalid.proofs.html invalid.proofs.html

Fair Use Books OnlineFair Use Books Online

http://fair-use.org/bertrand-russell/the-principles-of-http://fair-use.org/bertrand-russell/the-principles-of-mathematics/ mathematics/

BooksBooks

Dr. Euler's Fabulous Formula: Cures Many Mathematical Dr. Euler's Fabulous Formula: Cures Many Mathematical IllsIlls