Lec 6-7-8 DS

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Discrete Structures, ,

Muhammad Sultan ZiaAssistant Professor

COMSATS, Institute of Information Technology, Sahiwal

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Overview


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Predicate Logic

Propositional logic provides a useful setting in

which we can analyze many types of logicalargument

, ,propositional logic is inadequate. e.g.

All even numbers are integers. 8 is an even

number. Therefore 8 is an integer. It is not true that all prime numbers are odd.

Therefore there must be at least one prime number

that is not odd.COMSATS, Institute of Information Technology, Sahiwal

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Predicate Logic

There are situations (as in previous examples)

where propositional logic is inadequate,

because it cannot deal with the logical

u u watomic propositions

In order to analyze such arguments, we need to

look at the logical structure within atomicpropositions.

Predicate logic allows us to do thisCOMSATS, Institute of Information Technology, Sahiwal

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Predicate/Propositional Function

A predicate is a statement containing one or

more variables. If values are assigned to all the

variables in a predicate, the resulting statement

.Examples

x is a multiple of 5

P ( x ) = x < 5


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Functions with multiple variables:

P(x,y) = x + y == 0

P(1,2) is false, P(1,-1) is true

P(x,y,z) = x + y == z

P(3,4,5) is false, P(1,2,3) is true

P(x1,x

2,x

3 x

n) =


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Functions with multiple variables:

x is taller than y

a is greater than one of b, c

x is at least n inches taller than y


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Quantifiers

Why quantifiers?

Many things (in this course and beyond) are specifiedusing quantifiers

A quantifier is an operator that limits thevariables of a proposition

Two types:

Universal

Existential


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Universal Quantification

Represented by an upside-down A:

It means for all

Let P(x) = x+1 > x

We can state the following:

x P(x)

English translation: for all values of x, P(x) istrue

English translation: for all values of x, x+1>x istrue


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But is that always true?

x P(x) Let x = the character a

Is a+1 > a?

Let x = the province of Punjab Is Punjab+1 > Punjab?

You need to specify your universe! What values x can represent

Called the domain or universe of discourse by thetextbook


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Let the universe be the real numbers.

Let P(x) = x/2 < x Not true for the negative numbers!

, When the domain is all the real numbers

In order to prove that a universal quantification is true,it must be shown for ALL cases

In order to prove that a universal quantification is false,it must be shown to be false for only ONE case


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Given some propositional function P(x)

And values in the universe x1

.. xn

The universal quantification x P(x) implies:

P(x1) P(x

2) P(x

n)


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Existential Quantification

Represented by an backwards E:

It means there exists Let P(x) = x+1 > x

We can state the following: x P(x)

English translation: there exists (a value of) x

such that P(x) is true

English translation: for at least one value of x,x+1>x is true


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Note that you still have to specify your

universe

There is no numerical value x for which x+1

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Let P(x) = x+1 > x

There is a numerical value for which x+1>x

In fact, its true for all of the values of x!

Thus, x P(x) is true

In order to show an existential quantification istrue, you only have to find ONE value

In order to show an existential quantification isfalse, you have to show its false for ALLvalues


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Given some propositional function P(x)

And values in the universe x1

.. xn

The existential quantification x P(x) implies:

P(x1) P(x

2) P(x

n)


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A Note on Quantifiers Recall that P(x) is a propositional function

Let P(x) be x == 0

Recall that a proposition is a statement that iseither true or false

There are two ways to make a propositionalfunction into a proposition:

Supply it with a value

For example, P(5) is false, P(0) is true

Provide a quantification

For example, x P(x) is false and x P(x) is true

Let the universe of discourse be the real numbersCOMSATS, Institute of Information Technology, Sahiwal

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Binding Variables

Let P(x,y) be x > y

Consider: x P(x,y)

s s not a propos t on What is y?

If its 5, then x P(x,y) is false

If its x-1, then x P(x,y) is true

Note that y is not bound by a quantifierCOMSATS, Institute of Information Technology, Sahiwal

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Example 1

Write the following propositions in symbols:

For every number x there is a number y such that y= x + 1.

There is a number such that for ever number x

y = x + 1.

Solution:

Let P(x,y) denote the predicate y = x + 1.

Note carefully the difference in meaning between the two propositionsCOMSATS, Institute of Information Technology, Sahiwal

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Multivariate Quantification

When two or more variables are involved

each of which is bound by a quantifier, theorder of the binding is important and the

meaning often requires some thought.


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Quantifiers

When evaluating an expression such as

x y z P(x,y,z )

Translate the proposition in the same order to

ng s : There is anx such that for ally there is az

such thatP(x,y,z) holds.


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Parsing Example

P(x,y,z ) = y -x z

There is anx such that for ally there is az suchthaty -x z.

There is some number which when

subtracted from any number y results in anumber bigger than some numberz.

Q: If the universe of discourse forx, y, andz isthe non-negative integers {0,1,2,3,4,5,6,7,}whats the truth value ofxy z P(x,y,z )?


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Parsing ExampleA: True.

For any exists we need to find a positiveinstance.

Since x is the first variable in the expression and ,

all other y, z. Set x = 0 (want to ensure that y -xis not too small).

Now for each y we need to find a positive

instance z such that y - x z holds. Plugging in x= 0 we need to satisfy y z so we have manysolutions like z := y or z := 0 or some other validsolution.


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Parsing Example

Q: Isnt it simpler to satisfy

x y z (y -x z )

by setting x := y and z := 0 ?

A: No, this is illegal ! The existence of x comes

before we know about y. I.e., the scope of x is

higher than the scope of y so as far as y cantell, x is a constant and cannot affect x.


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Order Matters

Set the universe of discourse to be whole

numbers {0, 1, 2, 3, }. LetR (x,y ) = x

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Order Matters But Not Always

Q: What if we have two quantifiers of the same

kind? Does order still matter?No! If we have two quantifiers of the same kind

order is irrelevant.

x y is the same as y x because these areboth interpreted as for every combination ofxandy

x y is the same as y x because these areboth interpreted as there is a pairx ,y


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Translating from EnglishIn the specification of a system for booking theatre seats,

B(p,s) denotes the predicate person p has booked seat s.

Write the following sentences in symbolic form:

a) Seat s has been booked.

b) Person p has booked a (that is, at least one) seat.

c) All the seats are booked.

d) No seat is booked by more than one person.


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Translating from English

What about if the universe of discourse is all

students (or all people?) Every student in this class has studied calculus.

x (S(x)C(x))

This is wrong! Why?

x (S(x)C(x))


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Consider:

Some students have visited Mexico Every student in this class has visited Canada or

Mexico

Let:

S(x) be x is a student in this class

M(x) be x has visited Mexico C(x) be x has visited Canada


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Translating from English Consider: Some students have visited Mexico

Rephrasing: There exists a student who has visitedMexico

True if the universe of discourse is all students

What about if the universe of discourse is all people?

x (S(x) M(x)) This is wrong! Why?

x (S(x) M(x))


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Consider: Every student in this class has

visited Canada or Mexico

x x x

When the universe of discourse is all students

x (S(x)(M(x)C(x)) When the universe of discourse is all people


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Negating Quantifications Consider the statement:

All students in this class have black hair What is required to show the statement is false?

have black hair

To negate a universal quantification:

You negate the propositional function AND you change to an existential quantification

x P(x) = x P(x)


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Negating Quantifications

Consider the statement:

There is a student in this class with red hair

What is required to show the statement is false?

Thus, to negate an existential quantification:

To negate the propositional function

AND you change to a universal quantification x P(x) = x P(x)


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Negation Example

Compute: x y x2 y

In English, we are trying to find the opposite ofeveryx admits ay greater or equal toxs square.The opposite is that somex does not admit ay

Algebraically, one just flips all quantifiers from to and vice versa, and negates the interior

propositional function. In our case we get:

xy (x2 y ) xy x2 >y


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Translating between English and quantifiers

The product of two negative integers is positive

xy ((x0 >0 x+ /2 > 0

The difference of two negative integers is not necessarilynegative

xy ((x

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Translating between English and quantifiers

xy (x+y = y)

There exists an additive identity for all real numbers

xy (((x0) (y 0))

A non-ne ative number minus a ne ative number is reaterthan zero

xy (((x0) (y0)) (x-y > 0))

The difference between two non-positive numbers is notnecessarily non-positive (i.e. can be positive)

xy (((x0) (y0)) (xy 0))

The product of two non-zero numbers is non-zero if andonly if both factors are non-zero


Lec 6-7-8 DS

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