Math2305.Lecture 2 TAMUT

8/13/2019 Math2305.Lecture 2 TAMUT

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PREDICATES AND QUANTIFIERS

MATH-2305 Discrete Mathematics

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Truth Verification of those Statements

that are not Propositions

There are no rules in propositional logic thatallow to conduct the truth value of thefollowing statements:

There is at least one student in thisclassroom who took three Calculus classes

x+7=10

Each computer in this classroom is connectedto the network

y is a student

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Predicates

Statements like

x+7=10

y is a student

x 3have two parts.

The first part, a variable, is the subject of thestatement

The second part is the predicate. It refers to aproperty that the subject of the statement mayhave

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Predicates

Predicateis a dual-purpose term On the one hand, a predicateis a property or

description of subjects. The following predicatesare all shown in red and bold:

James is tall. The bridge is long.

19 is a prime number.

On the other hand, a predicate is also used as a

synonym of a propositional function, where thedescription is related not to a certain subject, butto a variable.


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Predicates and

Propositional Functions If a statement depends on a variable, this

statement is called a propositional function (oftena propositional function itself is called a

predicate) We can denote such a statement by .

Once a value has been assigned to the variable x,the statement becomes a proposition,

which has a truth value.

is also referred to as 1-place predicateorsimply a predicate

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P x

P x

P x


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Multivariable Propositional Functions

Multivariable propositional functionsdepend on

more than one variable. For example,

xis taller than y

ais greater than one ofb, c x is at leastninches taller than y

x+y=z

x andy are students


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n-ary Predicates

A multivariable propositional function depends

on more than one variable. For example, the

propositional function x+y=zdepends on 3

variables.

Once some values have been assigned to the

variable x, y, z, the statement x+y=z

becomes a proposition, which has a truth value. A statement of the form is called a

n-place predicateor n-ary predicate.

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1 2, ,..., nP x x x


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Propositional Functions and

Programming

Propositional functions are widely used incomputer programming and algorithm design forbranching programs and algorithms

If then

else

If (((x>0) and (xz))then

else

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Precondition and Postcondition

Preconditionis a statement that describes a valid

input of some algorithm or program segment

Postconditionis a statement that the output of

some algorithm or program segment should

satisfy when the algorithm (the program

segment) has run

Preconditions and Postconditions are widely usedto verify the correctness of algorithms and

programs

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Precondition and Postcondition

Example. Let we need to swap the values of

two variables x and y.

Algorithm: temp := x ; x := y; y := temp

Precondition: (x=a) & (y=b)

Postcondition: (x=b) & (y=a)

:= means set tox := y means set x equal to y

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Semantics

If logical propositions are to have meaning, they needto describe something. Up to now, propositions suchas Johnny is tall, Debbie is 5 years old, andx2=-1had no intrinsic meaning. Either they are true

or false, but no more. In order to make such propositions and propositional

functions meaningful, we need to have a domain(or universe) of discourse, (or simply domain(or universe)) i.e. a collection of subjectsabout which

the propositions relate.

Question: What are the domains for the threepropositions above?


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Semantics

Question: What are the domains for the followingpropositions and propositional functions?

Johnny is tall, Debbie is 5 years old

Possible answers: {Johnny, Debbie}; {People in theworld}; {People leaving in Texarkana}, etc.

x2=-1

Possible answers: Cthe set of complex numbers;Rthe set of real numbers, Zthe set of integernumbers

Depending on the particular domain, a propositionmay be true or false


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Semantics

Semantics is very important in computerprogramming

Each data structure and even each single variable

that are used in a computer program always havetheir domain of discourse (type).

For example, a variable xcan be declared as realor integer or byte or char.

Depending on this declaration, such expressionsas x=3.5, x=5, x=Class, x=-1 can be meaningfuland acceptable or not.

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Quantifiers

There are statements, which assert that someproperty is true for all values of a variable in aparticular domain (Each student in thisclassroom takes the Discrete Math course in Fall

semester 2012). There are also statements, which assert that

there is an element in a particular domain with acertain property (There is at least one person in

this classroom who is not a student). Such statements can be formulated using

quantifiers

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Quantifiers

There are two quantifiers

Universal Quantifier

reads for all or Each or Every Existential Quantifier

reads there exists or there is at leastone

A quantifier is placed in front of apropositional function and bindsit to obtain aproposition with semantic value.


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Quantifiers

A statement (propositional function) P(x) is quantifiedif there is a quantifier in a front of it, which is appliedto it:

x P(x) is TRUEif P(x) is true for every single x.

x P(x) is FALSEif there is an x for which P(x) is false.

x P(x) is TRUEif there is an x for which P(x) is true.

x P(x) is FALSEif P(x) is false for every single x.

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x P X x P X


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Quantifiers

The truth table for quantifiers

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The Truth Table for Quantifiers

Statement When True? When False?

P(x) is true

for every x

There is an x for

which P(x) is false

There is an x for

which P(x) is true

P(x) is false

for every x x P X

x P X


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The Universal Quantifier

x P (x)is true when everyinstance ofxmakesP (x) true when plugged in

Like conjunctioning over entire domain of P (x)

x P (x) P (x1) P (x2) P (x3)

Example: Each non-negative real number has a

square root


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Existential Quantifier

x P (x)is true when an instance can be found

which when plugged in forxmakes P (x) true

Like disjunctioning over entire domain of P (x)

x P (x) P (x1) P (x2) P (x3)

Example: There exist a complex number whose

square is a negative real number (i2=-1, iis an imaginary unity)


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Multivariate Quantification

Quantification involving only one variable is

fairly straightforward. Just a bunch of ORs or

a bunch of ANDs.

When two or more variables are involved each

of which is bound by a quantifier, the order of

the binding is important and the meaning

often requires some thought.


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Parsing Example

Question: If the domain forx, y, and z is the natural

numbers {0,1,2,3,4,5,6,7,} whats the truth

value of xy z P (x,y,z ); P (x,y,z ) = y - x z ?

Answer: True. For any existswe need to find a positive

instance. Sincex is the first variable in the expression and

is existential, we need a number that works forall other y, z. Setx= 0 (want to ensure that y -xis not too small).

Now for each y we need to find a positiveinstance z such thaty-x z holds. Plugging in

x= 0 we need to satisfy y z so set z = y.


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Parsing Example

Question: If the domain forx, y, and z is the natural

numbers {0,1,2,3,4,5,6,7,} , did we have to set z

= y to ensure that xy z P (x,y,z );

P (x,y,z ) = y - x z is true?

Answer: No. Could also have used the constant

z= 0. There are other valid solutions.


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Parsing Example

Question: Isnt it simpler to satisfy x y z (y-x z )

by settingx = y and z= 0 ?

Answer: No, this is illegal ! The existence ofx comes

before we know about y. I.e., the scopeofx is

higherthan the scopeof y . So we have to find first

suchx

that does not depend ony. Thus, it is illegal

to choosexdepending on y.


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Order matters

Set the domain of discourse to be all realnumbers .

Let P (x,y ) = x < y.

Question: What does x y P (x,y )mean? Answer: x y P (x,y ):

All numbersx admit a larger number y

Question: Whats the truth value of thisexpression?

Answer: True. For any real number there is alarger real number


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Order matters

P (x,y ) = x < y

Question: What does y x P (x,y ) mean?

Answer: y x P (x,y ):

Some number y is larger than allx

Question: Whats the truth value of this

expression?

Answer: False. There is no the largest real

number. Additionally the number cannot belarger than itself.


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Order mattersbut not always

If we have two quantifiers of the same kind,order does not matter.

x y is the same as y x because these

are both interpreted as for everycombination ofx and y

x y is the same as y x because these are

both interpreted asthere is a pairx , y


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Predicates - the meaning of multiple quantifiers

xy P(x,y)

xy P(x,y)

xy P(x,y)

xy P(x,y)

P(x,y) true for all (x, y) pairs.

For every value of x we can find a (possibly different)y so that P(x,y) is true.

P(x,y) true for at least one (x, y) pair.

There is at least one x for which P(x,y)

is always true.

Quantification order is not

commutative in

general !

Suppose P(x,y) = xsfavorite class is y.


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Negation of Logical Expressions

with Quantifiers

What about and

Since the quantifiers are the same as taking a bunch of

ANDs () or ORs () we obtain applying De Morganslaws:

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x P x ?x P x

1 2

1 2

...

.. ;

.

n

n

P x P x P x

P

x P x

x P xx P x P x

1 2

1 2

...

...

n

n

P x P x P x

P x P x

x P x

x P xP x


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Negation of Logical Expressions

with Quantifiers

General rule: to negate a logical expression

containing quantifier (quantifiers), move

negation to the expression under quantifier

(quantifiers) and flip all quantifiers from toand vice versa.

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Negation Example

Compute:

In English, we are trying to find the opposite ofeveryx admits a y greater or equal toxsquared.The opposite is that somex does not admit ygreater or equal toxsquared

Algebraically, one just flips all quantifiers fromto and vice versa, and negates the interiorpropositional function. In our case we get:

2x y x y

2 22x y x yx y yx x y xy


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Logical Equivalences

Involving Predicates and Quantifiers

Statements involving predicates and

quantifiers are logically equivalent if and only

if they have the same truth value no matter

which predicates are substituted into thesestatements and which domain is used for the

variables in these propositional functions.

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Math2305.Lecture 2 TAMUT

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