Discrete Mathematics Transparency No. 1-0 Chapter 1. The Foundations Cheng-Chia Chen.

Discrete Mathematics

Transparency No. 1-0

Chapter 1. The Foundations

Cheng-Chia Chen

The Foundations


1. The foundations

Logic: The basis of all mathematical reasoning has practical applications to the design of

computing machine give the precise meaning of math. statements.

Sets: The basis of all mathematical structures. definitions, operations

Functions : definitions, classification, operations

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1.1 LogicProposition:

A statement that is either true or false (,but not both).Example:1. Taipei is the capital of ROC. 2. Tokyo is the capital of Korea.3. 1+1 = 2. 4. 2+2 = 3.5. There is no integer x,y,z and n > 2 s.t.

xn + yn = zn ---- Fermat’s last theorem6. “Every even number is the sum of two prime numbers.” ---- Goldaach’s conjecture (1742)==>1. Statement (1)~(5) are propositions,2. (1),(3) are true; (2),(4) are false.3. Nobody knew (5) is true or false until 1994, but it must be true or false,

hence it is a proposition. (6) open problem.

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More Examples

Examples:1. What time is it? (interrogative)

2. Read this carefully! (command)

3. x+1 = 2.

4. x+y = z.

==> 1. (1) and (2) are not statements

2. (3) and (4) are not propositions:

neither true nor false!

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Propositional variables and propositional constants In algebra, we use variables to denote an item with un

specified value. e.g. 10 chickens and rabbits in a cage with 16 feet. ==> #rab

bits = ? and #chickens=? ==> let x = #rabbits, y = #chickens ==> x + y = 10 /\ 2x + 4y = 16.

Likewise, In logic, we can use propositional variable to denote an arbitrary proposition with unspecified truth value. e.g., If it is raining, the ground is wet. since the ground is w

et, it is raining now. ==> Let p =def “It is raining”, q =def “the ground is wet” ==> ((p ->q) /\ q) -> p.

Propositional constants: only two values: T(true), F(false).

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Forming Compound propositions from simpler ones Primitive propositions:

propositional variables propositional constants

P, Q: two known propositions Then the followings are also propositions:

1. “P and Q” : ( P /\ Q) conjunction of P and Q2. “It is not the case that p”: (~P) negation of P3. “P or Q” : (P \/ Q) disjunction of P and Q4. “P implies Q” : ( P Q) implication

P : antecedent, premise, hypotheses; Q: consequence, conclusion equivalent phrase:

– “P entails Q”, “Q if P”, “P only if Q”

5. “p iff q” (p <->q) equivalence of P and Q

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Truth conditions for compound propositions

1. P /\ Q is true iff both P and Q are true

2. P \/ Q is true iff either P or Q is true

3. ~P is true iff P is not true (i.e., false)

4. P<->Q is true iff both are true or both are false.

5. PQ is true iff whenever P is true, Q is true

( i.e., it is not the case that P is true but Q is false ) In the propositions, /\, \/, <-> and are called binary

connectives(or operations) and ~ is called unary connective(or operations).

Other binary connectives are possible.

What about the truth condition for primitive propositions?

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From natural language statements to formal logic expressions

Natural language statementToday is FridayI have a test todayIt is not the case that today is

FridayToday is not FridayToday is Friday and

I have a test todayIf today is Friday, then I have

a test today.Today is Friday or I have a

test today.

symbolized logic expr.pq~p

~p(P q)∧

(pq)

(P q)∨

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Truth conditions for propositions

ProblemProblem: : when will you say that the sentence:

It_is_raining is true ?

=> The proposition:” It_is_raining” is true if the meaning (or fact) that the proposition is intended to represent occurs(happens, exists) in the situation that the sentence referred to.

=>Example: Since it is not raining now(the current situation), the statement It_is_raining is false (in the current situation). But if it were raining now, then I would say that It_is_raining is true.

Factors affecting the truth value of a proposition: the situation in which the proposition is used. the meaning of the proposition.

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Truth table

A B ~A A/\B A\/B A->B A<->B

0 0 1 0 0 1 1

0 1 1 0 1 1 0

1 0 0 0 1 0 0

1 1 0 1 1 1 1

•The meaning of logical connectives can be represented by a truth table.

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Other usual connectives

Besides /\,\/, and <->, there are also some other connectives as you have known in digital system.

xor , nand, nor.Rules: let I(x) denote the truth value of x.

I(A B)= I(~(A<->B))=1 iff not I(A)=I(B) I(A nand B) = I(~(A/\B)). I(A nor B) = I(~( A \/B)).

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1.2 Propositional Equivalence

Some definitions: Let A be any proposition, then

1. A is a tautology(contradiction) iff A is true (false) no matter what the truth values of the prop. variables inside A are given.

2. If A is neither a tautology nor a contradiction, then A is called a contingency.

The geography of propositions

Tautologies(valid)

Contradictions(unsatisfiable)

Contingencies

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Logical Equivalence

A and B are logically equivalent (,A B) if A <-> B is a tautology

Theorem1: Logical equivalence relation is reflexive, symmetric and transitive.

I.e., for all propositions A,B and C, 1. A A2. A B implies B A and3. A B and B C imply A C.

Theorem2[substitution theorem]: If A B and C[X] is a proposition containing X as a subproposition, then C[A] and C[B] are logically equivalent, where C[A] is the result of C with X in C replaced by A. ex: (p q) ∨ (q p), C[X] =∨ def ~(p X) ∧ => ~(p (p q) )∧ ∨ ~(p (q p))∧ ∨

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Determine tautologies, contradictions and contingencies

Problems: Given a proposition A, how can you determine whether it is a tautology, a contradiction or a contingency?

=>Exhausted evaluation! (by using truth table)

e.g., A = (p->q) <-> (~p \/q) a tautology ?

1. there are two primitive props: p, q.

2. So there are 4 possible assignments of truth values to p and q.

p=0,q=0: p=0,q=1: p=1,q=0: p=1,q=1:

3. Under each assignment, we can evaluate the truth value of A (in bottom up way) by the truth condition rule of connecting connectives.

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Determine tautologies

p:0q:0

p->q:1

~p:1

~p \/q:1

(pq)<->~p\/q :1

The evaluation tree for (p->q)<-> (~p\/q)

p q ~p p->q ~p\/q p->q <-> ~p\/q

0 0 1 1 1 1

0 1

1 0

1 1

T

The truth table for pq <-> ~p\/q

11

1

1

00

0

1

1

1

01

A tautology since all are 1’s

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Determine other properties

1. logical equivalence: A <=> B iff A<->B is a tautology.

So we can apply the procedure for determining tautology to determine if A <=>B.

2. contradiction: A is a contradiction iff ~A is a tautology

3. Satisfiability: A is satisfiable iff ~A is not a tautology.

4. contingence: A is a contingence iff neither A nor ~A is a tautology.

5. logical consequence: B is a logical consequence of A [denoted A |= B] iff AB is a tautology.

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P /\ T P P/\P P P \/ F P P\/P PP \/ T P /\ F ~(~P) P \/ Q P /\ Q P \/ (Q \/ R) P /\ (Q /\ R) ==> P /\ (Q /\ R) = (P/\Q/\R) P /\(Q \/ R) P \/(Q /\ R) ~(P /\Q) ~(P\/Q) ==> ~(P1 /\ P1/\.../\Pn)

Some important logical equivalences

• Identity law

• Domination law

• Double negation

•Commutative

•Associative

•Distribution law

•DeMorgan’s law

•Idempotent law

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One Example:

Show that ~(p \/ (~p /\ q) ~p /\ ~q

Proof:1. By truth table (omitted)2. ~(p \/ (~p /\ q) deM

~p /\ ~(~p /\ q) deM

~p /\ (~(~p) \/ ~q) DbNeg

~p /\ (p \/ ~q) Distr

(~p \/ ~q) /\ (~p \/ p) ExMd

(~p \/ ~q) \/ T identity

(~p \/ ~q)

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1.3 Predicates (述詞 ) and quantifiers(量詞 )

Problems of Propositional logic: Cannot analyze the content of primitive propos

itions. cannot group similar propositions cannot analyze quantifiers.

Ex: Consider the following situations:

1. Family = {a,b,c1,c2,c3 } 2. We are interested in who is whose brot

her. 3. Suppose In Family, ci is cj’s brother if i j for i,j =1..3.

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Inadequancy of propositional logic

Primitive propositions needed: x_is_y’s_brother, x,y Family So we need 25 propositions.

New approach: Parameterized proposition(Predicate). Use a (predicate) symbol B to symbolize the

statement: B(x,y) x is y ‘s brother. e.g., B(a,c1) : a is c1’s brother, B(c2,c2) : c2 is c2’s brother, ...

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Benefits of Predicates

Advantage of using predicates:==>

1. need only 6 (1 predicate and 5 names) symbols.

2. can group related propositions together. (B(x,y) and others, for instance,

q(x) x_is_old. )3.The most important : allow representation of

quantified statement like: c1_has_a_brother (or, there is an object x which is c1’s brother, x B(x, c1))

to be inferred from

c2_is_c1’s brother ( B(c2,c1) )

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PredicatesIn the expression: B(x,y )

B : predicate symbol (or proposition function) x (y) : first (2nd) argument B(x,y) : [atomic] proposition(or formula) #arguments in B(X,Y): the arity of the predicate

symbol B.Notes:1. Each arity has a fixed arity.

2. B(x,y,c1) and B(x) are meaningless.

because used #arguments different from B’s arity.

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Predicates (cont’d)

3. The truth value of a proposition p(x1,...,xn) depends on both p and x1,...,xn.Ex1: Let p(x) “x >3 “. then

P(4) “4 > 3” has truth value T, P(2) “2 > 3” has truth value F.

Ex2: Let Q(x,y) “x = y +3” then Q(1,2) = “1 = 2 + 3” is false. Q(3,0) = “3 = 0 + 3” is true.

Mathematically, we can define (the meaning of) P as a function [P] [P]: Un (n is P’s arity) --> {T,F}with the intention that “P(x1,...,xn)” holds (is true) iff [P]([x1],...,[xn]) = T.

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Set representations of truth-functionsEx:

[B]: Family2 -->{T,F} s.t. [b](x,y) = T iff x =[ci], y=[cj], i j. p(x) “x >3 “ , So [P]: N --> {T,F } s.t. [P](x) = T iff x > 3. Q(x,y) “x = y +3” , so [Q] : N2 --> {T,F} with [Q](x,y) = T iff x = y +3.

Note: P partition N into two sets, one with members mapped to T and one mapped to F.

N

{4,5,...}{1,2,3}

P(x) = T.

P(x) = F

So, instead of representP as a truth function, we equivalently represent Pas the set {x | P(x) = T }= {4,5,6,...}

Similarly, we can represent Q as {(x,y) | x = y + 3 }

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constant, function, variables and terms

In the statement: “y > min(x, 3)” --- (s1) x,y are called variables, 3 is a constant, min is a function symbol with arity 2 “min(3,2)” behaves more like x, 3 than “x >y”. So if let P(x,y) “x > y”, then s1 can be represented as

P(y, min(x,3))we call any expression that can be put on the argu

ment position of an atomic proposition a termObviously, constants and variables are terms; mor

eover,if f is a function of arity n, and t1,...tn are n terms, then so is f(t1,...,tn).

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Universal and existential quantificationsA: any statement (or called formula)

e.g., A = p(x,y) /\ q(x,f(3)),...

x: any variable, Then 1. (x A) is a new formula called the universal

quantification of A, 2. (x A) means:

“A(x) is true for all values of x in the universe of discourse.”

3. (x A) is a new formula called the existential quantification of A,

4. (x A) means:“A(x) is true for some value of x in the universe of

discourse.”

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Quantifiers and universe of discourse

Consider the sentence: p(x) = “x > 0”. 1. as stated before, p(x) is true or false depending on

the value of x. 2. p(1), p(2),... : true; p(0) , p(-1),...: false

Universe of Discourse (U,論域 ): The set of objects (domain) from which all variables

are allowed to have values.Ex: consider the sentence:

s2: ”all numbers are greater than 0”.

(for-all x “x > 0” ) or ( x p(x) ) U = N+={1,2,3,...} => s2 is true. U = Z={...,-1,0,1,...}=> s

2 is false.

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Translation of NL statement into logical one

Translate the following sentences into logical expressions:

1. Every body likes Chang2. Everybody loves somebody.3. There is somebody loved by every one.4. Nobody likes everybody.5.There is somebody Chang don’t like.6. there is somebody no one likes.7.there is exactly one person everybody likes.8 Chang likes exactly two persons9. Everyone likes himself.10. There is someone who likes no one who does not

like himself.

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Terminology about formulasOccurrences of variables in formulas:

A = p(f(x), c) \/ q(x,y) A contains 2 variables, one of which occurs twic

e.Free and bound variables:A: a formulafv(A) = the set of all free variables in A is defined as

follows: If A is atomic => fv(A) is the set of all variables i

n A fv(~A) = fv(A) fv(A/\B)=fv(A\/B)=fv(A->B)=fv(A<->B)= fv(A) U fv

(B). fv( for-all x A) = fv(xA) = fv(A)\ {x}.

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bd(A): the set of bound variables in A is defined as follows; 1. If A is atomic => bd(A) = {} 2. bd(~A) = bd(A) 3. bd(A/\B) = d(A\/B) = bd(AB) =bd(A<->B) = bd(A) U bd(B). 4. bd( x A) = ∀

bd(A) if x ∉ fv(A). bd(A) U {x} if x fv(A). ∈Example A = ( x p(x,y)) \/ q(x) => fv(A) = {x,y}; bd(A) = {x} x occurs twice in A, one occurs free and the other occur

s bound.Def: A is a sentence iff fv(A) = {}

Note: A is a proposition if A is a sentence

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More logical notions [omitted!!]A B [logical equivalence (in predicate logic)] iff

A <-> B is a tautology(or valid) for all possible values of free variables in A and in B.

Common equivalences:1. all propositional equivalences

A->B ~A \/ B, ~~A A, ~(A/\B) ~A\/~B,...

2. ~ x A < x ~A. 3. ~xA x ~A. 4. x (A /\ B) x A /\ xB. 5. x(A \/B) xA \/ XB If x is not free in B, and Q is or , then 6 Qx (A /\ B) QxA /\ QxB 7. Qx(A\/B) QxA \/ QxB

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1.4 Sets

Basic structure upon which all other (discrete and continuous ) structures are built.

a set is a collection of objects. an object is anything of interest, maybe itself a set.

Definition 1. A set is a collection of objects. The objects is a set are called the elements or me

mbers of the set. If x is a member of a set S, we say S contains x. notation: x S vs x S

Ex: In 1,2,3,4,5, the collection of 1,3 5 is a set.

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Set descriptionHow to describe a set:?1. List all its member.

the set of all positive odd integer >10 = ? The set all decimal digits = ? the set of all upper case English letters = ? The set of all nonnegative integers = ?

2. Set builder notation: P(x) : a property (or a statement or a proposition)

about objects. e.g., P(x) = “ x > 0 and x is odd” then {x | P(x) } is the set of objects satisfying

property P. P(3) is true => 3 {x | P(x)} P(2) is false => 2 {x | P(x)}

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Conventions

N =def {x | x is a natural numbers } = { 0,1,2,3,...}

N+ =def {1,2,3,...}

Z =def {...,-3,-2,-1,0,1,2,3,...}

R =def the set of real numbers.

Problem:

The same set may have many different descriptions. {x | 0 < x <10 /\ x is odd } {1,3,5,7,9}, {5,3,1,9,7} {9,7,1,3,5}.

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Set predicatesDefinition 2.

Two sets S1, S2 are equal iff they have the same elements

S1 = S2 iff x (x S1 <-> x S2)Ex: {1,3,5} = {1,5,3} = {1,1,3,3, 5}Graphical representation of sets:

Venn Diagrams: rectangle: Universal set: U circles: sets points: particular elements point x inside circle S => x S point x outside S => x S.

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Set predicates (cont’d)Null set ={} = =def the collection of no objects.Def 3’: [empty set] for-all x x.Def 3. [subset]

A B iff all elements of A are elements of B. A B for-all x (x A x ⇒ B)).

Def 3’’: A B =def A B /\ A B.Exercise : Show that:

1. For all set A ( 2. (A B /\ B A) (A = B) 3. A ⇒

Diagram representation of the set inclusion relationship.

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Size or cardinality of a set

Def. 4: | A | = the size(cardinality) of A = # of distinct elements of A.

Ex: |{1,3,3,5}| = ? |{}| = ? | the set of binary digits } | = ? |N| = ? ; |Z| = ? ; | {2i | i N} = ? ∈ |R| = ?

Def. 5. A set A is finite iff |A| is a natural number ; o/w it is

infinite. Two sets are of the same size (cardinality) iff there

is a 1-1 & onto mapping between them.

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countability of sets

Exercise: Show that 1. |N| = |Z | = | Q | = |{4,5,6,...}| 2. |R| = | (-1, 1) | = |(0,1)| 3. |(0,1)| is uncountable

Def. A set A is said to be denumerable iff |A| = |N|. A set is countable iff either |A| = n for some n i

n N or |A| = |N|.

By exercise 1,2,3, R is not countable. Q and Z is countable.

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The power set

Def 6. If A is a set, then the collection of all subsets of A

is also a set, called the poser set of A and is denoted as P(A) or 2A.

Ex: P({0,1,2}) = ? P({}) = ? |P({1,2,..., n})| = ?

Order of elements in a set are indistinguishable. But some sometimes we need to distinguish between (1,3,4) and (3,4,1) --> ordered n-tuples

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Show that |A| ≠ |2A|

Pf: (1) The case that A is finite is trivial since |2A| = 2|A

| > |A| and there is no bijection b/t two finite sets with diffe

rent sizes.(2) assume |A| = |2A|, i.e., there is a bijection f: A -> 2A. Let D = {x A | x ∈ f(x) }. ==> 1. D is a subset of A; Hence 2. y A s.t. f(y) = D.∈ Problem: Is y D ? if yes (i.e., y D) ==> y f(y) = D, a contradiction if no (i.e., y D) ==> y f(y) =D, a contradiction too. So the assumption is false, i.e., there is no bijection

b/t A and 2A.

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The Halting ProblemL : any of your favorite programming languages (C,

C++, Java, BASIC, etc. )Problem: write an L-program HALT(P,X), which takes

another L-program P(-) and string X as input, and

HALT(P,X) returns true if P(X) halts and

returns false if P(X) does not halt.

P(X) Halt ?

yes true

no false

Pr(x:String) {…..

}

P

“It’s a test …”

X

HALT(P,X)

Program you are asked to

design

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Halt(P,X) does not exist

Motivations of the proof:Problem1 : What about the truth value of the sentence:

L: L is false Problem 2 : Let S = {X | X X}. Does S belong to S or no

t ?The analysis: S S => S S; S S => S S.Conclusion:

1. S is not a set!! 2. If a language is too powerful, it may produce expressions

that is meaningless or can not be realized.Question: If HALT(P,X) can be programmed, will it inc

ur any absurd result like the case of S?Ans: yes!!

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H(P) : Another program using HALT(-,-) as a subroutine

H(P)

Pr(x:String) {…..

}

P

P(P) Halt ?

yestrue

nofalse

HALT(P,X) Loop

END

Note: H(P) Halts iff HALT(P,P) returns false iff P(P) does not halt.

P

P

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P(X) Halt ?

yes true

no false

HALT(P,X)

H(P)

Loop

END

P(X) Halt ?

yestrue

nofalse

HALT(P,X)

H(H(P))

Loop

END

Problem: Will H(H(P)) Halt?

H(P)

The anomaly :H(H(P) halts iff H(H(P)) does not halt.

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Cartesian Products

Def. 7 [n-tuple] If a1,a2,...,an (n > 0) are n objects, then “(a1,a2,...,an)

” is a new object, called an (ordered) n-tuple [ with ai as its ith element. ]

Any ordered 2-tuple is called a pair. (a1,a2,...,am) = (b1,b2,...,bn) iff

(1) m = n and (2) for i = 1,..,n ai = bi.

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Cartesian product

Def. 8: [Cartesian product]

A x B =def {(a,b) | a A and b B }∈ ∈ A1 x A2 x ...x An =def {(a1,...,an) | ai Ai }.∈

Ex: A = {1,2}, B = {a,b,c} , C = {0,1}

1. A x B = ? ; 2. B x A = ?

3. A x {} = ? ;4. A x B x C = ?

problem: 1. when will A x B = B x A ?

2. |A x B | = ?

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1.5 Set operations

union, intersection,difference , complement,Definition. 1. A B = {x | x A or x B }∈ ∈ 2. A B = {x | x A and x B }∈ ∈ 3. If A B = {} => call A and B disjoint. 4. A - B = {x | x A but x ∈ ∉ B } 5. ~ A = U - AVenn diagram representationsEx: U = {1,..,10}, A = { 1,2,3,5,8} B = {2,4,6,8,10}=> A B , A B , A - B , ~ A = ?

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Set identities

Identity laws: A ? ? = ADomination law: U ? ? = A; {} ?? = {}Idempotent law: A ? A = A ;complementation: ~~A = A

commutative : A ? B = B ? AAssociative: A ? (B ? C) = (A ? B ) ? CDistributive: A ? (B ? C) = ?DeMoregan laws: ~(AUB) = ?; ~(A ⋂ B)=?

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Prove set equality1. Show that ~(A B) = ~A ~B by show that

1. ~(A B)~A ~B 2. ~A ~B ~(A B)

pf: (By definition) Let x be any element in ~(A B)

2. show (1) by using set builder and logical equivalence.

3. Show distributive law by using membership table.

4. show ~(A (B C)) = (~C ~B) ~A by set identities.

Note the similarity between logical equivalence and set identities.

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Generalized set operations

Def. 6 A1,A2,...An: n sets B = {A1,A2,...An }

n{i=1,..n} Ai =def ?

n{i=1,..n} Ai=def ?

quiz: if B = {} => = ?; Venn diagram representation of

andExample: = ?

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Set representation (in computer)

Unordered list or array union, intersection, ~: time consuming.

Bit string: U = {a1,...,an} is the universal set. A: any subset of U A can be represented by the string:

s(A) = x1 x2 x3....xn where

xi = 1 if a1 A and 0 o/w.∈ fast set operations suitable only if U is not large. constant size representation.

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1.6 FunctionsDef. 1 [functions] A, B: two sets

1. a function f from A to B is a set of pairs (x, y) AxB ∈s.t., for each x A there is at most one y B s.t. (x,y) ∈ ∈

f.∈ 2. if (x,y) f, we write f(x) = y.∈ 3. f :A B means f is a function from A to B.Def. 2. If f:A B then

1. A: the domain of f; 2. B: the codomain of f if f(a)=b => 3. b is the image of a 4. a is the preimage of b 5. range(f) = ? 6. preimage(f) = ? 7. f is total iff ?

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Types of functionsDef 4. f: A x B; S: a subset of A,

T: a subset of B

1. f(S) =def ?

2. f-1(T) =def ?Def. [1-1, onto, injection, surjection, bijection] f: A -> B.

f is 1-1 (a injection) iff ? f is onto (surjective, a surjection) iff ? f is 1-1 & onto <=> f is bijective (a bijection, 1-1 co

rrespondence) Exercise:Show that if there are onto mappings from A

to B and from B to A, then |A|=|B|.

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Real valued functions

F:A B is a real valued function iff A and B are subsets of R.

f1,f2: real valued functions

=> 1. f1+f2 (x) = ?

2. f1• f2 (x) = f1(x) x f2(x).

3. f is increasing iff ?

4. f is strictly increasing iff ?

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operations on functions

A, B, C: any sets ; f: A B; g: B C, then

1. [identity function]

idA : A A s.t. idA(x) = x for all x in A.

2. [composition of g and f]

gf: A C s.t. gf(x) = g(f(x)) for x in A.

3. [inverse] If f is a bijection, then

f-1: B A s.t. f-1(y) = x iff f(x) = y for y in B. Graphical representations

The Foundations


Properties of operations on functions

f: A B. g: B C; h: C D, then 1. idA f = f = f idB

2. f(gh) = (fg) h --- assoc.

3. (fg)-1 = g-1 f-1

If f: A A is a bijection then

4. f f-1 = f-1 f= idA .

Sequence: finite A-sequence: a: [n] (or [1,n]) A A-sequence: a : N (or N+) A. B indexed A-sequence : f:BA.

The Foundations


The growth of functions

Summation rules ai =def a1 + a2+...an+ (...) a + (a+d) + (a+2d)+... = ? a + ar +arr + ... = ?

The growth of functions:If f is a positive valued function, then what d

o the following terms mean? O(f) means ? o(f) means ? (f) means ? (f) means ? (f) means ? useful in the analysis of the efficiency of comput

er algorithms.

The Foundations


Problem, algorithm and Complexity

A problem is a general question: description of parameters [input] description of solution[output]

An algorithm is a step by step procedure to solve a problem. a recipe a computer program

We want the most efficient algorithm fastest (mostly) most economical with memory (sometimes) expressed as a function of problem size

The Foundations


Parameters: Set of cities Inter-city distances

Example: Traveling Salesman Problem

c

bd

a

3

5

6

9

910

The Foundations


Solution [output]

Solution: The shortest tour passing all cities Example: a,b,d,c,a has length 27

c

bd

a

3

5

6

9

910

The Foundations


Problem Size

What is appropriate measure of problem size? m nodes? m(m+1)/2 distances?

Use an encoding of the problem alphabet of symbols: a,b,c,d,0-9, |. strings: abcd||10|5|9|6|9|3.

Measures Problem Size: length of encoding. Time Complexity: how long an algorithm takes,

as function of problem size.

The Foundations


Time Complexity

What is tractable? A function f(n) is O(g(n)) whenever ∃ c > 0 n∃ 0 > 0 s.t. |f(n)| ≤ c ∙ |g(n)| for all n > n0.

A polynomial time algorithm is one whose time complexity is O(p(n)) for some polynomial p(n).

An exponential time algorithm is one whose time complexity cannot be bounded by a polynomial

(e.g., nlog n ).

The Foundations


Tractability Basic distinction:

polynomial time = tractable exponential time = intractable

10 20 30 40 50 60

n .00001s .00002s .00003s .00004s .00005s .00006s

n2 .0001s .0004s .0009s .0016s .0025s .0036s

n3 .001s .008s .027s .064s .125s .216s

n5 .1s 3.2s 24.3s 1.7 m 5.2 m 13.0 m

2n .001s 1.0s 17.9s 12.7d 35.7 y 366c

3n .059 s 58 m 6.5 y 3855 c 2∙108c 1.3∙10 13 c

execution time in microseconds(μs)

The Foundations


Effect of Speed-ups for different order of functionsWait for faster hardware! Consider maximum problem size you can solve in an

hour. present 100 times faster 1000 times faster

n N1 100 N1 1000 N1

n2 N2 10N2 31.6 N2

n3 N3 4.64 N3 10 N3

n5 N4 2.5 N4 3.98 N4

2n N5 N5+6.64 N5+9.97

3n N6 N6+4.19 N6+2.29

The Foundations


Asymptotic notations

((g(n)) = {f(n) | c2,c1 and n0 > 0 s.t. c2g(n) f(n) c1 g(n) for all n n0 }

((g(n)) = {f(n) | c and n0 > 0 s.t. f(n) cg(n) for all n n0 }

((g(n)) = {f(n) | c and n0 > 0 s.t. c g(n) f(n) for all n n0 }

((g(n)) = {f(n) | arbitrary small c >0 n0 > 0 s.t. f(n) < c g(n) for all n n0 }

((g(n)) = {f(n) | arbitrary large c >0 n0 > 0 s.t. c g(n) < f(n) for all n n0 }

The Foundations


Using the asymptotic notations

We use f(n) = (g(n)) to mean f(n) (g(n)), where = ,or .

analog: O (asymptotic upper bound): f(n) = O(g(n)) means like f(n) g(n) o (asymptotic upper bound but not tight): < f(n) = o(g(n)) means like f(n) < g(n) (asymptotic lower bound): f(n) =(g(n)) means like f(n) g(n) (asymptotic lower bound but not tight): > f(n) = (g(n)) means like f(n) > g(n) Q (asymptotic tight bound : = f(n) = O(g(n)) means like f(n) = g(n)

The Foundations


Exercises

Let f and g be positive real function. Show that

1. f = (g) iff g = (f)

2. f =(g) iff g = (f)

3. f =o(g) iff f =O(g) but not f = (g)

4 f = (g) iff f =(g) but not f = (g)

5 f = o(g) iff limit n f/g = 0 iff

limit n g/f = iff g = (f).

Discrete Mathematics Transparency No. 1-0 Chapter 1. The Foundations Cheng-Chia Chen.

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Transcript of Discrete Mathematics Transparency No. 1-0 Chapter 1. The Foundations Cheng-Chia Chen.