2015 discrete mathematics ch1.1-1.3 0226

8/18/2019 2015 Discrete Mathematics CH1.1-1.3 0226

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Discrete Mathematics

離散數學

江振國


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課程內容

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單主題容綱要

1. The Foundations: Logic

and Proofs

Propositional logic, applications of propositional

logic, propositional equivalences, predicates and

quantifiers , rules of Inference

2. Basic Structures: Sets,

Functions, SequencesBasic definitions and operations

3. Algorithms Time complexity, tractable and intractable

4. Number Theory andCryptography

Divisibility and modular arithmetic,integer representations and algorithms,

5. Induction and

Recursion

Mathematical induction, strong induction and well-

ordering, recursive definitions and structural

induction


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課程內容

3

單主題容綱要

6. Counting The basics of counting, the pigeonhole principle

7. Discrete Probability Probability theory, Bayes’ theorem

8. Advanced Counting Applications of recurrence relations, solving linearrecurrence relations

9.RelationsProperties of relations, representing relations,

equivalence relation and closures of relations .

10.GraphsBasic definitions and properties, shortest paths,

Euler paths, Hamiltonian paths

11.TreesBasic definitions and properties, prefix codes,

binary search trees, minimum spanning trees


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課程評量

• 評分方式• 上課點名 10%

• 小考 15%, 作業 25%

• 期中考 25%, 期末考 25%

• 教科書• Discrete Mathematics and Its Applications, K. H. Rosen

• Reference: Elements of discrete mathematics, C.L. Liu,McGraw-Hill

• 東華書局 – 謝韡韜0927887330, [email protected]

4


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Chapter Summary

• Propositional Logic• The Language of Propositions

• Applications

• Logical Equivalences

• Predicate Logic• The Language of Quantifiers

• Logical Equivalences

• Nested Quantifiers

• Proofs• Rules of Inference

• Proof Methods

• Proof Strategy


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Propositional Logic Summary

• The Language of Propositions• Connectives

• Truth Values

• Truth Tables

• Applications• Translating English Sentences

• System Specifications

• Logic Puzzles

• Logic Circuits

• Logical Equivalences• Important Equivalences• Showing Equivalence

• Satisfiability


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Propositional Logic

Section 1.1


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Propositions

• A proposition is a declarative sentence that is either trueor false.

• Examples of propositions:

a) The Moon is made of green cheese.

b) Toronto is the capital of Canada.

c) 1 + 0 = 1

d) 0 + 0 = 2

• Examples that are not propositions.a) Sit down!

b) What time is it?

c) x + 1 = 2

d) x + y = z


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Propositional Logic

• Constructing Propositions• Propositional Variables: p, q, r , s, …

• The proposition that is always true is denoted by T andthe proposition that is always false is denoted by F.

• Compound Propositions; constructed from logicalconnectives and other propositions

Negation ¬

Conjunction ∧

Disjunction ∨

Implication →

Biconditional ↔


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Compound Propositions: Negation

• The negation of a proposition p is denoted by ¬ p and has this truth table:

• Example: If p denotes “The earth is round.”, then

¬ p denotes “It is not the case that the earth is

round,” or more simply “The earth is not round.”

p ¬pT F

F T


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Conjunction

• The conjunction of propositions p and q is denoted by p ∧ q and has this truth table:

• Example: If p denotes “I am at home.” and q denotes “It is raining.” then p ∧q denotes “I am athome and it is raining.”

p q p ∧ q

T T T

T F F

F T F

F F F


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Disjunction

• The disjunction of propositions p and q isdenoted by p ∨q and has this truth table:

• Example: If p denotes “I am at home.” and q denotes “It is raining.” then p ∨q denotes “Iam at home or it is raining.”

p q p ∨q

T T T

T F T

F T T

F F F


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The Connective Or in English

• In English “or” has two distinct meanings.• “Inclusive Or” - In the sentence “Students who have taken CS202 or

Math120 may take this class,” we assume that students need to have taken one

of the prerequisites, but may have taken both. This is the meaning of

disjunction. For p ∨q to be true, either one or both of p and q must be true.

• “Exclusive Or” - When reading the sentence “Soup or salad comes with this

entrée,” we do not expect to be able to get both soup and salad. This is the

meaning of Exclusive Or (Xor). In p ⊕ q , one of p and q must be true, but not

both. The truth table for ⊕ is:

p q p ⊕q

T T F

T F T

F T T

F F F


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Implication

• If p and q are propositions, then p →q is a conditional statement or implication which is read as “if p , then q ”and has this truth table:

• Example: If p denotes “I am at home.” and q denotes “Itis raining.” then p →q denotes “If I am at home then it israining.”

• In p →q , p is the hypothesis (antecedent or premise) andq is the conclusion (or consequence).

p q p →q

T T T

T F F

F T T

F F T


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Understanding Implication

• One way to view the logical conditional is to think ofan obligation or contract.

• “If I am elected, then I will lower taxes.”

• “If you get 100% on the final, then you will get an A.”

• If the politician is elected and does not lower taxes,then the voters can say that he or she has broken the

campaign pledge. Something similar holds for the professor. This corresponds to the case where p istrue and q is false.


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Understanding Implication (cont)

• In p →q there does not need to be any connection between the antecedent or the consequent. The

“meaning” of p →q depends only on the truth

values of p and q .• These implications are perfectly fine, but would

not be used in ordinary English.

• “If the moon is made of green cheese, then I havemore money than Bill Gates. ”

• “If 1 + 1 = 3, then your grandma wears combat boots.”


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Different Ways of Expressing p →q

• if p, then q

• if p, q

• q unless ¬p

• q if p

• q whenever p

•q follows from p

• a necessary condition for p is q

• a sufficient condition for q is p

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• p implies q

• p only if q

• q when p

• p is sufficient for q

• q is necessary for p


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Converse, Contrapositive, and Inverse

• From p →q we can form new conditional statements .• q → p is the converse of p →q

• ¬q → ¬ p is the contrapositive of p →q

• ¬ p → ¬ q is the inverse of p →q

Example: Find the converse, inverse, andcontrapositive of “If it is raining, I’m not going totown.”

Solution:converse: If I do not go to town, then it is raining.

contrapositive: If I go to town, then it is not raining.

inverse: If it is not raining, then I will go to town.


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Biconditional

• If p and q are propositions, then we can form thebiconditional proposition p ↔q , read as “ p if and only if

q .” The biconditional p ↔q denotes the proposition with

this truth table:

• If p enotes “You can take the flight.” and q denotes

“You buy a ticket.” then p ↔q denotes “You can take

the flight if and only if you buy a ticket.”

p q p ↔q

T T T

T F F

F T F

F F T


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Expressing the Biconditional

• Some alternative ways “ p if and only if q” isexpressed in English:

• p is necessary and sufficient for q

• if p then q , and if q then p

• p iff q


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Truth Tables For Compound Propositions

• Construction of a truth table:

• Rows

• Need a row for every possible combination of values for

the atomic propositions.• Columns

• Need a column for the truth value of each expression that

occurs in the compound proposition as it is built up.

This includes the atomic propositions

• Need a column for the compound proposition (usually at

far right)


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Example Truth Table

• Construct a truth table for

p q r p q r p → r

T T T T F FT T F T T T

T F T T F F

T F F T T T

F T T T F F

F T F T T T

F F T F F T

F F F F T T


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Equivalent Propositions

• Two propositions are equivalent if they alwayshave the same truth value.

• Example: Show using a truth table that the

implication is equivalent to the contrapositive.Solution:

p q

¬

p

¬

q p →q

¬

q → ¬ p

T T F F T T

T F F T F F

F T T F T T

F F T T T T

i h bl Sh


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Using a Truth Table to Show

Non-Equivalence

Example: Show using truth tables that neither theconverse nor inverse of an implication are not

equivalent to the implication.

Solution:

p q ¬ p ¬ q p →q ¬ p →¬ q q → p

T T F F T T T

T F F T F T T

F T T F T F F

F F T T T T T


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Precedence of Logical Operators

Operator Precedence

1

2

3

4

5

p q r is equivalent to ( p q ) r

If the intended meaning is p ( q r )

then parentheses must be used.


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Applications of Propositional

LogicSection 1.2

A li i f P i i l L i


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Applications of Propositional Logic:

Summary

• Translating English to Propositional Logic

• System Specifications

• Boolean Searching

• Logic Puzzles

• Logic Circuits

• AI Diagnosis Method (Optional)


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Translating English Sentences

• Steps to convert an English sentence to astatement in propositional logic

• Identify atomic propositions and represent using propositional variables.

• Determine appropriate logical connectives

• “If I go to Harry’s or to the country, I will not goshopping.”

• p: I go to Harry’s• q: I go to the country.

• r : I will go shopping.

If p or q then not r .


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Example

Problem: Translate the following sentence into propositional logic:

“You can access the Internet from campus only if

you are a computer science major or you are not afreshman.”

One Solution: Let a , c , and f represent respectively

“You can access the internet from campus,” “Youare a computer science major,” and “You are a

freshman.” a→ (c ∨ ¬ f )


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System Specifications

• System and Software engineers take requirements inEnglish and express them in a precise specificationlanguage based on logic.

Example: Express in propositional logic:

“The automated reply cannot be sent when the filesystem is full”

Solution: One possible solution: Let p denote “The

automated reply can be sent” and q denote “The filesystem is full.”

q→ ¬ p


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Consistent System SpecificationsDefinition: A list of propositions is consistent if it

is possible to assign truth values (true or false) to

the proposition variables so that each proposition

is true.

Exercise: Are these specifications consistent?

“The diagnostic message is stored in the buffer or

it is retransmitted.”

“The diagnostic message is not stored in the buffer.”

“If the diagnostic message is stored in the buffer,

then it is retransmitted.”


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Logic Puzzles

• An island has two kinds of inhabitants, knights,

who always tell the truth, and knaves, who always

lie.

• You go to the island and meet A and B.

• A says “B is a knight.”

• B says “The two of us are of opposite types.”

Example: What are the types of A and B?


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Logic Puzzles

• Solution: Let p and q be the statements that A is aknight and B is a knight, respectively. So, then p

represents the proposition that A is a knave and

q that B is a knave.• If A is a knight, then p is true. Since knights tell the

truth, q must also be true. Then B says (p ∧ q)∨ (

p ∧ q) would have to be true, but it is not. So, A is not

a knight and therefore p must be true.

• If A is a knave, then B must not be a knight since

knaves always lie. So, then both p and q hold since

both are knaves. 37


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Logic Circuits

• Electronic circuits; each input/output signal can be viewed as a 0 or 1.• 0 represents False

• 1 represents True

• Complicated circuits are constructed from three basic circuits called gates.

• The inverter (NOT gate)takes an input bit and produces the negation of that bit.

• The OR gate takes two input bits and produces the value equivalent to the

disjunction of the two bits.

• The AND gate takes two input bits and produces the value equivalent to the

conjunction of the two bits.

• More complicated digital circuits :


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Propositional Equivalences

Section 1.3


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Section Summary

• Tautologies, Contradictions, and Contingencies.

• Logical Equivalence

• Important Logical Equivalences

• Showing Logical Equivalence

• Propositional Satisfiability

• Sudoku Example

Tautologies Contradictions and


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Tautologies, Contradictions, and

Contingencies

• A tautology is a proposition which is always true.• Example: p ∨¬ p

• A contradiction is a proposition which is always

false.• Example: p ∧¬ p

• A contingency is a proposition which is neither a

tautology nor a contradiction, such as p ¬p p ∨¬p p ∧¬p

T F T F

F T T F


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De Morgan’s Laws

p q ¬p ¬q ( p ∨q ) ¬( p ∨q ) ¬p ∧¬ q

T T F F T F F

T F F T T F F

F T T F T F F

F F T T F T T

This truth table shows that De Morgan’s Second Law holds.

Augustus De Morgan

1806-1871


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Key Logical Equivalences

• Identity Laws: ,

• Domination Laws: ,

• Idempotent laws: ,

• Double Negation Law:

• Negation Laws: ,


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Key Logical Equivalences (cont )

• Commutative Laws: ,

• Associative Laws:

• Distributive Laws:

• Absorption Laws:

M L i l E i l


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More Logical Equivalences


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More Logical Equivalences


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Constructing New Logical Equivalences

• We can show that two expressions arelogically equivalent by developing a series

of logically equivalent statements.

• To prove that we produce a series

of equivalences beginning with A and ending

with B.


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Equivalence Proofs

Example: Show thatis a tautology.

Solution:

q

by negation law


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Propositional Satisfiability

• A compound proposition is satisfiable if there isan assignment of truth values to its variables that

make it true. When no such assignments exist, the

compound proposition is unsatisfiable.• A compound proposition is unsatisfiable if and

only if its negation is a tautology.


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Questions on Propositional Satisfiability

Example: Determine the satisfiability of the followingcompound propositions:

Solution: Satisfiable. Assign T to p, q, and r .

Solution: Satisfiable. Assign T to p and to q .

Solution: Not satisfiable. Check each possible assignmentof truth values to the propositional variables and none willmake the proposition true.


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Notation

Needed for the next example.


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Sudoku

• A Sudoku puzzle is represented by a 99 gridmade up of nine 33 subgrids, known as blocks.

Some of the 81 cells of the puzzle are assigned

one of the numbers 1,2, …, 9.• The puzzle is solved by assigning numbers to each

blank cell so that every row, column and block

contains each of the nine possible numbers.• Example


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Encoding as a Satisfiability Problem

• Let p(i, j,n) denote the proposition that is truewhen the number n is in the cell in the ith row and

the jth column.

• There are 999 = 729 such propositions.• In the sample puzzle p(5,1,6) is true, but p(5, j,6) is

false for j = 2,3,…9


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Encoding (cont)

• For each cell with a given value, assert p(i, j,n),when the cell in row i and column j has the given

value.

• Assert that every row contains every number.

• Assert that every column contains every number.


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Encoding (cont)

• Assert that each of the 3 x 3 blocks contain everynumber.

(this is tricky - ideas from chapter 4 help)

• Assert that no cell contains more than one

number. Take the conjunction over all values of n,

n’ , i, and j, where each variable ranges from 1 to 9and of

l i i fi bili bl


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Solving Satisfiability Problems

• To solve a Sudoku puzzle, we need to find an assignmentof truth values to the 729 variables of the form p(i,j,n) that

makes the conjunction of the assertions true. Those

variables that are assigned T yield a solution to the puzzle.

• A truth table can always be used to determine the

satisfiability of a compound proposition. But this is too

complex even for modern computers for large problems.

• There has been much work on developing efficientmethods for solving satisfiability problems as many

practical problems can be translated into satisfiability

problems.

2015 discrete mathematics ch1.1-1.3 0226

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