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Mathematics for computer science

conf.dr. Bostan Viorel

Spring 2013 Lecture 1

conf.dr. Bostan Viorel () MathDisc Spring 2013 Lecture 1 1 / 35

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Mathematical Logic

1 Stefan Bilaniuk, A problem course in mathematical logic.

2 Eric Lehman, F.Thomson Leighton, Albert R. Meyer, Math forcomputer science, MIT course.

3 Victor Besliu, Matematica discreta, UTM.


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Mathematical Logic

Denition

Mathematical logic is concerned with formalizing and analyzing the kindsof reasoning used in mathematics.


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Mathematical Logic

Denition


Part of the problem with formalizing mathematical reasoning is thenecessity of precisely specifying the language(s) in which it is to be done.


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Mathematical Logic

Denition


Part of the problem with formalizing mathematical reasoning is thenecessity of precisely specifying the language(s) in which it is to be done.

The natural languages spoken by humans wont do: they are so complexand continually changing as to be impossible to pin down completely.


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Mathematical Logic. Some history

EntscheidungsproblemGiven a set of hypotheses and some statement , is there an eectivemethod for determining whether or not the hypotheses in are sucientto prove ?


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Historically, this question arose out of David Hilberts scheme to securethe foundations of mathematics by axiomatizing mathematics in 1st-orderlogic, showing that the axioms in question do not give rise to anycontradictions, and that they suce to prove or disprove every statement

(which is where the Entscheidungsproblem comes in).


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Historically, this question arose out of David Hilberts scheme to securethe foundations of mathematics by axiomatizing mathematics in 1st-orderlogic, showing that the axioms in question do not give rise to anycontradictions, and that they suce to prove or disprove every statement

(which is where the Entscheidungsproblem comes in).

If the answer to the Entscheidungsproblem were "yes" in general, theeective method(s) in question might put mathematicians out of business.


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David Hilbert (1862-1943)


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Trying to nd a suitable formalization of the notion of"eective

method", mathematicians developed abstract models of computation inthe 1930s: recursive functions, calculus, Turing machines, andgrammars.


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These models are very dierent,


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These models are very dierent, but they were all essentially equivalent in

what they could do.conf.dr. Bostan Viorel () MathDisc Spring 2013 Lecture 1 5 / 35

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This suggested the (empirical, not mathematical!) principle:


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Church-Turing ThesisA function is eectively computable in principle in the real world if and onlyif it is computable by (any) one of the abstract models mentioned above.


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Church-Turing ThesisA function is eectively computable in principle in the real world if and onlyif it is computable by (any) one of the abstract models mentioned above.

Alonzo Church(1903-1995)

Alan Turing(1912-1954)


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Mathematical Logic

Consider some sequences taken from English language:

1 "You may have cake or you may have ice cream."


M h i l L i

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Mathematical Logic



2 "If you dont clean your room, then you wont play Counter Strike!"


M h i l L i

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Mathematical Logic




3 "If pigs can y, then you can understand the Banach-TarskyTheorem."


M h i l L i

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Mathematical Logic





4 "If you can solve any problem we come up with in this class, then youget grade 10 for this course."


M th ti l L i

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Mathematical Logic





4 "If you can solve any problem we come up with in this class, then youget grade 10 for this course."

5 "Every human has a dream."


M th ti l L i

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Mathematical Logic

"You may have cake or you may have ice cream."


Mathematical Logic

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Mathematical Logic


Can you have both cake and ice cream or must you choose just one desert?


Mathematical Logic

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Mathematical Logic



"If you dont clean your room, then you wont play Counter Strike!"


Mathematical Logic

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Mathematical Logic




If you arent playing the game, does it mean that you didnt clean yourroom?


Mathematical Logic

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Mathematical Logic





"If pigs can y, then you can understand the Banach-Tarsky Theorem."


Mathematical Logic

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Mathematical Logic





"If pigs can y, then you can understand the Banach-Tarsky Theorem."

If this is true, then is the Banach-Tarsky Theorem incomprehensible?


Mathematical Logic

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Mathematical Logic

"If you can solve any problem we come up with in this class, then you getgrade 10 for this course."


Mathematical Logic

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Mathematical Logic


If you can solve some problems we come up with but not all, then do you

get a 10 for the course?


Mathematical Logic

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Mathematical Logic



get a 10 for the course?And can you still get a 10 even if you cant solve any of the problems?


Mathematical Logic

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Mathematical Logic




"Every human has a dream."


Mathematical Logic

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Mathematical Logic




"Every human has a dream."

Does the last sentence imply that all humans have the same dream ormight they each have a dierent dream?


Mathematical Logic

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g

Some uncertainty is tolerable in normal conversation. But ...


Mathematical Logic

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g


Imperativ!We need to formulate ideas precisely as in mathematics.


Mathematical Logic

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g


Imperativ!We need to formulate ideas precisely as in mathematics. The ambiguitiesinherent in everyday language become a real problem!


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g



We cant hope to make an exact argument if were not sure exactly whatthe individual words mean.


Mathematical Logic

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To get around the ambiguity of English, mathematicians have devised aspecial language for talking about logical relationships.


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This language mostly uses ordinary English words and phrases such as


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This language mostly uses ordinary English words and phrases such as"or",


Mathematical Logic

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This language mostly uses ordinary English words and phrases such as"or", "implies",


Mathematical Logic

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This language mostly uses ordinary English words and phrases such as"or", "implies", and "for all".


Mathematical Logic

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This language mostly uses ordinary English words and phrases such as"or", "implies", and "for all".

But mathematicians endow these words with denitionsmore precisethanthose found in an ordinary dictionary.


Propositional Logic

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Propositional logic formalizes the reasoning that can be done with

connectivessuch as not, and, or, and if . . . then.


Propositional Logic

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Dene the formal language of propositional logic, LPby specifying itssymbols and rules for assembling these symbols into the formulas of thelanguage.


Propositional Logic

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Dene the formal language of propositional logic, LPby specifying itssymbols and rules for assembling these symbols into the formulas of thelanguage.

Denition

The symbols ofLP are:

1 Parentheses: ( and )

2 Connectives: : and !

3 Atomic formulas: A0, A1, A2,. . ., An,. . .


Propositional Logic

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How to put together the symbols ofLP?


Propositional Logic

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Denition

The formulas ofLPare those nite sequences or strings of the symbolsgiven in previous denition which satisfy the following rules:

1 Every atomic formula is a formula;2 If is a formula, then (:) is a formula;

3 If and are formulas, then (!) is a formula;

4 No other sequence of symbols is a formula.


Propositional Logic

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Denition

The formulas ofLPare those nite sequences or strings of the symbolsgiven in previous denition which satisfy the following rules:

1 Every atomic formula is a formula;2 If is a formula, then (:) is a formula;

3 If and are formulas, then (!) is a formula;

4 No other sequence of symbols is a formula.

Use lower-case Greek characters to represent formulas, and upper-caseGreek characters to represent sets of formulas.


Propositional Logic

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A0 = "The moon is red"


Propositional Logic

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A1 = "The moon is made of cheese"


Propositional Logic

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Then

(A0 !(:A1))

means


Propositional Logic

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Then

(A0 !(:A1))

meansIf the moon is red, then it is not made of cheese!


Propositional Logic

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Then

(A0 !(:A1))


These are formulas:


Propositional Logic

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Then

(A0 !(:A1))


These are formulas:A2013,


Propositional Logic

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Then

(A0 !(:A1))


These are formulas:A2013,(A100 !A1),


Propositional Logic

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Then

(A0 !(:A1))


These are formulas:A2013,(A100 !A1),(A0 !A0),


Propositional Logic

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Then

(A0 !(:A1))


These are formulas:A2013,(A100 !A1),(A0 !A0),((:A1)!(A2 !A231 ))


Propositional Logic

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Then

(A0 !(:A1))



These are NOT formulas:


Propositional Logic

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Then

(A0 !(:A1))



These are NOT formulas:X2,


Propositional Logic

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Then

(A0 !(:A1))



These are NOT formulas:X2,(A3),


Propositional Logic

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Then

(A0 !(:A1))



These are NOT formulas:X2,(A3),(A0 !(:A1),


Propositional Logic

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Then

(A0 !(:A1))



These are NOT formulas:X2,(A3),(A0 !(:A1),(A7:A1),


Propositional Logic

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Then

(A0 !(:A1))



These are NOT formulas:X2,(A3),(A0 !(:A1),(A7:A1),A2 !A0


Propositional Logic

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Use the symbols ^, _, and $ to represent and, or, and if and only if,respectively.


Propositional Logic

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Since they are not among the symbols ofLP , we will use them asabbreviations for certain constructions involving only : : and ! .


Propositional Logic

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Namely,


Propositional Logic

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Namely,

( ^) is short for (:(!(:)))


Propositional Logic

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Namely,

( ^) is short for (:(!(:)))

( _) is short for ((:)!)


Propositional Logic

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Namely,

( ^) is short for (:(!(:)))


($) is short for ((!) ^ (!))


Propositional Logic

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Namely,

( ^) is short for (:(!(:)))


($) is short for ((!) ^ (!))


Propositional Logic

U h b l d d d if d l if

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Namely,

( ^) is short for (:(!(:)))


($) is short for ((!) ^ (!))

"The moon is red and made of cheese" is written as(A0 ^ A1).


Propositional Logic

U h b l ^ _ d d d if d l if

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Namely,

( ^) is short for (:(!(:)))


($) is short for ((!) ^ (!))

"The moon is red and made of cheese" is written as(A0 ^ A1).

Or actually is (:(A0 !(:A1)))


Propositional Logic

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For simplicity adapt informal conventions (that will allow us to use fewer

parentheses):

Drop the outermost parentheses in a formula, writing ! insteadof(!) and : instead of(:);


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parentheses):


Let : take precedence over ! when parentheses are missing, so:! is short for ((:)! ), and t the informal connectivesinto this scheme by letting the order of precedence be:

:, ^, _, !, $;


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parentheses):


Let : take precedence over ! when parentheses are missing, so:! is short for ((:)! ), and t the informal connectivesinto this scheme by letting the order of precedence be:

:, ^, _, !, $;

Group repetitions of!, ^, _, or$ to the right when parenthesesare missing, so !! is short for ((!)!).


Propositional Logic

Denition

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Denition

Suppose is a formula ofLP. The set of subformulas of, S(), is

dened as follows:

1 If is an atomic formula, then S() =fg ;

2 If is (:), then S() =S() [ f:g ;

3 If is (!), then S() =S() [ S() [ f(! )g .


Propositional Logic

Denition

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Denition


dened as follows:


2 If is (:), then S() =S() [ f:g ;

3 If is (!), then S() =S() [ S() [ f(! )g .

For example, let be the formula

(((:A0)!A1)!(A2 !(:A1)))


Propositional Logic

Denition

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dened as follows:


2 If is (:), then S() =S() [ f:g ;

3 If is (!), then S() =S() [ S() [ f(! )g .

For example, let be the formula

(((:A0)!A1)!(A2 !(:A1)))

Then

S() =fA0, A1, A2,(:A0),((:A0)!A1),(:A1),(A2 !(:A1)),g


Propositional Logic

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Observe that dropping parentheses convention allow us to rewrite

(((:A0)!A1)!(A2 !(:A1)))

in


Propositional Logic

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(((:A0)!A1)!(A2 !(:A1)))

in(:A0 !A1)!(A2 ! :A1)


Propositional Logic

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(((:A0)!A1)!(A2 !(:A1)))

in(:A0 !A1)!(A2 ! :A1)

and

S() =fA0, A1, A2,(:A0),((:A0)!A1),(:A1),(A2 !(:A1)),g


Propositional Logic

O

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(((:A0)!A1)!(A2 !(:A1)))

in(:A0 !A1)!(A2 ! :A1)

and

S() =fA0, A1, A2,(:A0),((:A0)!A1),(:A1),(A2 !(:A1)),g

can be rewritten as

S() =fA0, A1, A2, :A0, :A0 !A1, :A1, A2 ! :A1, g


Propositional Logic

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:A0 ^ :A1 $ :(A0 _ A1)


Propositional Logic

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:A0 ^ :A1 $ :(A0 _ A1)

Using parentheses it should be

(((:A0) ^ (:A1)) $(:(A0 _ A1)))


Propositional Logic

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:A0 ^ :A1 $ :(A0 _ A1)

Using parentheses it should be

(((:A0) ^ (:A1)) $(:(A0 _ A1)))

The subformullas are

S() =fA0, A1, :A0, :A1, :A0 ^ :A1, A0 _ A1, :(A0 _ A1), g

conf dr Bostan Viorel () MathDisc Spring 2013 Lecture 1 18 / 35

Propositional Logic

Whether a given formula ofLP is true or false usually depends on howwe interpret the atomic formulas which appear in

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we interpret the atomic formulas which appear in .


Propositional Logic


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If=fA2g and A2 = "2+2=4",


Propositional Logic


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If=fA2g and A2 = "2+2=4", then is True,


Propositional Logic


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If=fA2g and A2 = "2+2=4", then is True,but ifA2 = "The moon is made of cheese",


Propositional Logic


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If=fA2g and A2 = "2+2=4", then is True,but ifA2 = "The moon is made of cheese", it is False.


Propositional Logic

Whether a given formula ofLP is true or false usually depends on howwe interpret the atomic formulas which appear in .

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Not any statement can be assigned true or false value.


Propositional Logic


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A0 = "This statement is false"


Propositional Logic


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p pp




At this stage logical relationships are important.


Propositional Logic


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p pp





Lets dene how any assignment of truth values T ("true") and F ("false")

to atomic formulas ofLPcan be extended to all other formulas.


Propositional Logic


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p pp





Lets dene how any assignment of truth values T ("true") and F ("false")

to atomic formulas ofLPcan be extended to all other formulas.

We will also get a reasonable denition of what it means for a formula ofLPto follow logically from other formulas.


Propositional Logic

Denition

A truth assignment is a function v whose domain is the set of all formulas

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A truth assignment is a function vwhose domain is the set of all formulas

ofLPand whose range is the set fT; Fg of truth values, such that:


Propositional Logic

Denition


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ofLPand whose range is the set fT; Fg of truth values, such that:1 v(An ) is dened for every atomic formula An.


Propositional Logic

Denition


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2 For any formula ,

v((:)) = T,

ifv

() =F,

F, ifv() =T,

conf dr Bostan Viorel () MathDisc Spring 2013 Lect re 1 20 / 35

Propositional Logic

Denition


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2 For any formula ,

v((:)) = T, ifv

() =F,

F, ifv() =T,

3 For any formulas and ,

v((!)) = F, ifv() =T and v() =F

T, otherwise,

f d B t Vi l () M thDi S i 2013 L t 1 20 / 35

Propositional Logic

Denition


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2 For any formula ,

v((:)) = T, ifv(

) =F,

F, ifv() =T,


v((!)) = F, ifv() =T and v() =F

T, otherwise,


Propositional Logic

Denition


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g


2 For any formula ,

v((:)) = T, ifv() =F,

F, ifv() =T,


v((!)) = F, ifv() =T and v() =F

T, otherwise,

Truth assignment of implication, !, means that T !F is false.


Propositional Logic

Example.

Supposevis a truth assignment such that v(A0) = T and v(A1) =F.

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Propositional Logic

Example.


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Then we want to know the truth assignmemnt

v(((:A1)!(A0 !A1)))


Propositional Logic

Example.


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v(((:A1)!(A0 !A1)))

A0 A1 (:A1) (A0 !A1) ((:A1)!(A0 !A1))T F T F F


Propositional Logic

Example.


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v(((:A1)!(A0 !A1)))

A0 A1 (:A1) (A0 !A1) ((:A1)!(A0 !A1))T F T F F

In other words we showed that ifv(A0) =T and v(A1) =F, then


Propositional Logic

Example.


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v(((:A1)!(A0 !A1)))

A0 A1 (:A1) (A0 !A1) ((:A1)!(A0 !A1))T F T F F


v(((:A1)!(A0 !A1))) =F


Propositional Logic

Example.


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v(((:A1)!(A0 !A1)))

A0 A1 (:A1) (A0 !A1) ((:A1)!(A0 !A1))T F T F F


v(((:A1)!(A0 !A1))) =F

What if another truth asignment is given?


Propositional Logic

Example.


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v(((:A1)!(A0 !A1)))

A0 A1 (:A1) (A0 !A1) ((:A1)!(A0 !A1))T F T F F


v(((:A1)!(A0 !A1))) =F

What if another truth asignment is given? Say v(A0) =F and v(A1) =F.


Propositional Logic

Construct the so-called truth table with all possible truth asignments:

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Propositional Logic


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A0 A1 (:A1) (A0 !A1) ((:A1)!(A0 !A1))

T T F T T

T F T F F

F T F T T

F F T T T


Propositional Logic


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A0 A1 (:A1) (A0 !A1) ((:A1)!(A0 !A1))

T T F T T

T F T F F

F T F T T

F F T T T

Clearly, if there are three atomic formulas present, then we will have 8possible dierent truth asignments.


Propositional Logic


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A0 A1 (:A1) (A0 !A1) ((:A1)!(A0 !A1))

T T F T T

T F T F F

F T F T T

F F T T T


How about n atomic formulas?


Propositional Logic


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A0 A1 (:A1) (A0 !A1) ((:A1)!(A0 !A1))

T T F T T

T F T F F

F T F T T

F F T T T


How about n atomic formulas? Answer:


Propositional Logic


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A0 A1 (:A1) (A0 !A1) ((:A1)!(A0 !A1))

T T F T T

T F T F F

F T F T T

F F T T T


How about n atomic formulas? Answer: 2n possible truth asignments


Propositional Logic

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A0 A1 A2 . . . An1 An Subformulas

T T T . . . T T

T T T . . . T F

T T T . . . F F

. . . . . . . . . . . . . . . . . .

F F F . . . F T

F F F . . . F F


Propositional Logic

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PropositionSuppose is any formula and uand vare truth assignments such thatu(An ) =v(An ) for all atomic formulas An which occur in . Thenu() =v().


Propositional Logic

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PropositionSuppose is any formula and uand vare truth assignments such thatu(An ) =v(An ) for all atomic formulas An which occur in . Thenu() =v().

Corollary

Supposeuand vare truth assignments such that u(An ) =v(An ) for everyatomic formulaAn. Then u=v, i.e. u() =v() for every formula .


Propositional Logic

:

T F

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F T


Propositional Logic

:

T F

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F T

!

T T T

T F F

F T TF F T


Propositional Logic

:

T F

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F T

!

T T T

T F F

F T TF F T

^ :(!(:))

T T T T

T F F F

F T F F

F F F F


Propositional Logic

_

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T T T

T F T

F T T

F F F


Propositional Logic

_

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T T T

T F T

F T T

F F F

$

T T T

T F F

F T F

F F T


Propositional LogicDenition

Ifv is a truth assignment and is a formula, we will often say that vsatises ifv() =T . Similarly, if is a set of formulas, we will often

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say that v satises ifv() =T for every 2 . We will say that (respectively, ) is satisable if there is some truth assignment whichsatises it.




( )

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Denition

A formula is a tautology if it is satised by every truth assignment.




h f ( ) T f W ll h

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Denition


For example, ! is a tautology.




h i if ( ) T f W ill h

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Denition


For example, ! is a tautology.

DenitionA formula is a contradiction if there is no truth assignment whichsatises it.

For example, ! : is a contradiction.conf.dr. Bostan Viorel () MathDisc Spring 2013 Lecture 1 27 / 35

Propositional Logic

E l Sh h A3 (A4 A3) i l

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Example. Show that A3 !(A4 !A3) is a tautology.


Propositional Logic

E l Sh th t A3 (A4 A3) i t t l C t t th

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Example. Show that A3 !(A4 !A3) is a tautology. Construct thetruth table


Propositional Logic

E ample Sh th t A3 (A4 A3) i t t l C t t th

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Example. Show that A3 !(A4 !A3) is a tautology. Construct thetruth table

A3 A4 (A4 !A3) A3 !(A4 !A3)

T T T T

T F T T F T F T

F F T T


Propositional Logic

Proposition

If is any formula, then ((:) _ ) is a tautology and ((:) ^ ) is a

contradiction

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contradiction.


Propositional Logic

Proposition


contradiction

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contradiction.

Proof.

It follows from the truth tables


Propositional Logic

Proposition


contradiction

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contradiction.

Proof.


: (:) _ T F T

F T T


Propositional Logic

Proposition


contradiction

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contradiction.

Proof.


: (:) _ T F T

F T T

: (:) ^

T F FF T F



A proposition is a statement that is either true or false.

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Note that even if this denition is quite general it excludes such sentences

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Note that even if this denition is quite general it excludes such sentencesas

1 "O Romeo, Romeo! Wherefore art thou Romeo?"





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1 "O Romeo, Romeo! Wherefore art thou Romeo?"2 "Dont be stupid."





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3 "Learn mathematics!"





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Proposition

2 + 3 = 5.





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Proposition

2 + 3 = 5.

Proposition

Pigs can y.


Propositional LogicProposition A

All Greeks are human.

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Proposition B

All humans are mortal.




P i i B

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Proposition B


Proposition C

All Greeks are mortal.




P i i B

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Proposition B


Proposition C


Archimedes spent some time playing with such sentences in the 4th

century BC.




P iti B

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Proposition B


Proposition C


Archimedes spent some time playing with such sentences in the 4th

century BC.

If A is true, and B is true, then C is also true!


Propositional Logic

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Propositional Logic

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Archimedes developed an early form of logic.


Propositional LogicProposition

Let p(n) =n2 + n+41. Then 8n2 N, p(n) is a prime number.

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Propositional LogicProposition


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p(0) = 41 : prime


Propositional Logic

Proposition


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p(0) = 41 : prime

p(1) = 43 : prime


Propositional Logic

Proposition


( )

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p(0) = 41 : prime

p(1) = 43 : prime

p(2) = 47 : prime


Propositional Logic

Proposition


(0) 1 i

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p(0) = 41 : prime

p(1) = 43 : prime

p(2) = 47 : prime

p(3) = 57 : prime


Propositional Logic

Proposition


(0) 41 i

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p(0) = 41 : prime

p(1) = 43 : prime

p(2) = 47 : prime

p(3) = 57 : prime

So far so good!


Propositional Logic

Proposition


(0) 41 i

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p(0) = 41 : prime

p(1) = 43 : prime

p(2) = 47 : prime

p(3) = 57 : prime

So far so good!

p(20) =461 :prime!


Propositional Logic

Proposition


(0) 41 i

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p(0) = 41 : prime

p(1) = 43 : prime

p(2) = 47 : prime

p(3) = 57 : prime

So far so good!

p(20) =461 :prime!

Beautiful!

p(39) =1601 :prime!


Propositional Logic

Proposition


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Propositional Logic

Proposition


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p(40) =1681 :Not prime!

p(40) = 402+40+41

= 40(40+1) +41

= 40 41+41

= 41 41


Propositional Logic

Proposition


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p(40) =1681 :Not prime!

p(40) = 402+40+41

= 40(40+1) +41

= 40 41+41

= 41 41

Therefore, the above proposition is FALSE!


Propositional Logic

Proposition

"a4 + b4 + c4 =d4 has no solutions, where a, b, c and dare positiveintegers "

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Propositional Logic

Proposition


In logical notations (mathematicians like notations a lot!) it says

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8a2 Z+, 8b2 Z+, 8c2 Z+, 8d2 Z+, a4 + b4 + c4 6=d4.


Propositional Logic

Proposition



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8a2 Z+, 8b2 Z+, 8c2 Z+, 8d2 Z+, a4 + b4 + c4 6=d4.

8a, b, c, d2 Z+, a4 + b4 + c4 6=d4.


Propositional Logic

Proposition



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g ( ) y

8a2 Z+, 8b2 Z+, 8c2 Z+, 8d2 Z+, a4 + b4 + c4 6=d4.

8a, b, c, d2 Z+, a4 + b4 + c4 6=d4.

Euler made this conjecture in 1769. For 218 years no one knew whetherthis proposition was true or false.


Propositional Logic

Proposition



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g ( ) y

8a2 Z+, 8b2 Z+, 8c2 Z+, 8d2 Z+, a4 + b4 + c4 6=d4.

8a, b, c, d2 Z+, a4 + b4 + c4 6=d4.


Finally, Noam Elkies from Harvard University found a solution to theequation:

a=2682440, b=15365639, c=18796760, d=20615673.conf.dr. Bostan Viorel () MathDisc Spring 2013 Lecture 1 35 / 35

Propositional Logic

Proposition



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g ( ) y

8a2 Z+, 8b2 Z+, 8c2 Z+, 8d2 Z+, a4 + b4 + c4 6=d4.

8a, b, c, d2 Z+, a4 + b4 + c4 6=d4.


Finally, Noam Elkies from Harvard University found a solution to theequation:

a=2682440, b=15365639, c=18796760, d=20615673.conf.dr. Bostan Viorel () MathDisc Spring 2013 Lecture 1 35 / 35

MDSlides1_2013

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