Discrete Mathematics, Part II

CSE 2353

Fall 2007

Margaret H. DunhamMargaret H. DunhamDepartment of Computer Science and Department of Computer Science and

EngineeringEngineeringSouthern Methodist UniversitySouthern Methodist University

•Some slides provided by Dr. Eric Gossett; Bethel University; St. Paul, Some slides provided by Dr. Eric Gossett; Bethel University; St. Paul, MinnesotaMinnesota•Some slides are companion slides for Some slides are companion slides for Discrete Mathematical Discrete Mathematical Structures: Theory and Applications Structures: Theory and Applications by D.S. Malik and M.K. Senby D.S. Malik and M.K. Sen

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Outline

Introduction Sets Logic & Boolean Algebra Proof TechniquesProof Techniques Counting Principles Combinatorics Relations,Functions Graphs/Trees Boolean Functions, Circuits

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Proof Technique: Learning Objectives

Learn various proof techniques

Direct

Indirect

Contradiction

Induction

Practice writing proofs

CS: Why study proof techniques?

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Proof Techniques Theorem

Statement that can be shown to be true (under certain conditions)

Typically Stated in one of three ways

As Facts

As Implications

As Biimplications

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Validity of Arguments Proof: an argument or a proof of a theorem

consists of a finite sequence of statements ending in a conclusion

Argument: a finite sequence of statements.

The final statement, , is the conclusion, and the statements are the premises of the argument.

An argument is logically valid if the statement formula is a tautology.

AAAAA nn,...,,,,

1321

An

AAAA n 1321...,,,,

AAAAA nn

1321...,,,,

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Proof

A mathematical proof of the statement S is a sequence of logically valid statements that connect axioms, definitions, and other already validated statements into a demonstration of the correctness of S. The rules of logic and the axioms are agreedupon ahead of time. At a minimum, the axioms should be independent and consistent. The amount of detail presented should be appropriate for the intended audience.

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Proof Techniques

Direct Proof or Proof by Direct Method Proof of those theorems that can be expressed in

the form ∀x (P(x) → Q(x)), D is the domain of discourse

Select a particular, but arbitrarily chosen, member a of the domain D

Show that the statement P(a) → Q(a) is true. (Assume that P(a) is true

Show that Q(a) is true By the rule of Choose Method (Universal

Generalization), ∀x (P(x) → Q(x)) is true

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Proof Techniques Indirect Proof

The implication P → Q is equivalent to the implication ( Q → P)

Therefore, in order to show that P → Q is true, one can also show that the implication ( Q → P) is true

To show that ( Q → P) is true, assume that the negation of Q is true and prove that the negation of P is true

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Proof Techniques Proof by Contradiction

Assume that the conclusion is not true and then arrive at a contradiction

Example: Prove that there are infinitely many prime numbers

Proof: Assume there are not infinitely many prime numbers,

therefore they are listable, i.e. p1,p2,…,pn

Consider the number q = p1p2…pn+1. q is not divisible by any of the listed primes

Therefore, q is a prime. However, it was not listed. Contradiction! Therefore, there are infinitely many

primes.

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Proof Techniques Proof of Biimplications

To prove a theorem of the form ∀x (P(x) ↔ Q(x )), where D is the domain of the discourse, consider an arbitrary but fixed element a from D. For this a, prove that the biimplication P(a) ↔ Q(a) is true

The biimplication P ↔ Q is equivalent to (P → Q) ∧ (Q → P)

Prove that the implications P → Q and Q → P are true Assume that P is true and show that Q is true Assume that Q is true and show that P is true

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Proof Techniques

Proof of Equivalent Statements Consider the theorem that says that statements

P,Q and r are equivalent

Show that P → Q, Q → R and R → P Assume P and prove Q. Then assume Q and

prove R Finally, assume R and prove P

What other methods are possible?

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Other Proof Techniques

Vacuous

Trivial

Contrapositive

Counter Example

Divide into Cases

Constructive

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Proof Basics

You can not prove by example

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Proof Strategies with Quantifiers

Existential Constructive

some mathematicians only accept constructive proofs Nonconstructive

show that denying existence leads to a contradiction

Universal to prove false:

one counter-example to prove true:

usually harder the choose method

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Proofs in Computer Science

Proof of program correctness

Proofs are used to verify approaches

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Mathematical Induction Assume that when a domino is knocked over, the next domino is knocked over by it Show that if the first domino is knocked over, then all the dominoes will be knocked

over

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Mathematical Induction

Let P(n) denote the statement that then nth domino is knocked over

Base Step: Show that P(1) is true Inductive Hypothesis: Assume some P(i) is

true, i.e. the ith domino is knocked over for some

Inductive Step: Prove that P(i+1) is true, i.e.1i

)1()( iPiP

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Outline

Introduction Sets Logic & Boolean Algebra Proof Techniques Counting PrinciplesCounting Principles Combinatorics Relations,Functions Graphs/Trees Boolean Functions, Circuits

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Learning Objectives

Learn the basic counting principles—multiplication and addition

Explore the pigeonhole principle

Learn about permutations

Learn about combinations

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Basic Counting Principles

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Basic Counting Principles

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Pigeonhole Principle

The pigeonhole principle is also known as the Dirichlet drawer principle, or the shoebox principle.

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Pigeonhole Principle

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Permutations

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Permutations

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Combinations

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Combinations

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Generalized Permutations and Combinations