1 Predicates and Quantifiers CS/APMA 202, Spring 2005 Rosen, section 1.3 Aaron Bloomfield

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Predicates and Predicates and QuantifiersQuantifiers

CS/APMA 202, Spring 2005CS/APMA 202, Spring 2005

Rosen, section 1.3Rosen, section 1.3

Aaron BloomfieldAaron Bloomfield

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Terminology reviewTerminology review

Proposition: a statement that is either true Proposition: a statement that is either true or falseor false Must always be one or the other!Must always be one or the other! Example: “The sky is red”Example: “The sky is red” Not a proposition: x + 3 > 4Not a proposition: x + 3 > 4

Boolean variable: A variable (usually p, q, Boolean variable: A variable (usually p, q, r, etc.) that represents a propositionr, etc.) that represents a proposition

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Propositional functionsPropositional functions

Consider P(x) = x < 5Consider P(x) = x < 5 P(x) has no truth values (x is not given a P(x) has no truth values (x is not given a

value)value) P(1) is trueP(1) is true

The proposition 1<5 is trueThe proposition 1<5 is true P(10) is falseP(10) is false

The proposition 10<5 is falseThe proposition 10<5 is false

Thus, P(x) will create a proposition when Thus, P(x) will create a proposition when given a valuegiven a value

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Propositional functions 2Propositional functions 2

Let P(x) = “x is a multiple of 5”Let P(x) = “x is a multiple of 5” For what values of x is P(x) true?For what values of x is P(x) true?

Let P(x) = x+1 > xLet P(x) = x+1 > x For what values of x is P(x) true?For what values of x is P(x) true?

Let P(x) = x + 3Let P(x) = x + 3 For what values of x is P(x) true?For what values of x is P(x) true?

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Anatomy of a propositional functionAnatomy of a propositional function

P(x) = x + 5 > xP(x) = x + 5 > x

variable predicate

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Propositional functions 3Propositional functions 3

Functions with multiple variables:Functions with multiple variables:

P(x,y) = x + y == 0P(x,y) = x + y == 0P(1,2) is false, P(1,-1) is trueP(1,2) is false, P(1,-1) is true

P(x,y,z) = x + y == zP(x,y,z) = x + y == zP(3,4,5) is false, P(1,2,3) is trueP(3,4,5) is false, P(1,2,3) is true

P(xP(x11,x,x22,x,x33 … x … xnn) = …) = …

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So, why do we care about So, why do we care about quantifiers?quantifiers?

Many things (in this course and beyond) Many things (in this course and beyond) are specified using quantifiersare specified using quantifiers In some cases, it’s a more accurate way to In some cases, it’s a more accurate way to

describe things than Boolean propositionsdescribe things than Boolean propositions

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QuantifiersQuantifiers

A quantifier is “an operator that limits the A quantifier is “an operator that limits the variables of a proposition”variables of a proposition”

Two types:Two types: UniversalUniversal ExistentialExistential

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Universal quantifiers 1Universal quantifiers 1

Represented by an upside-down A: Represented by an upside-down A: It means “for all”It means “for all” Let P(x) = x+1 > xLet P(x) = x+1 > x

We can state the following:We can state the following: x P(x)x P(x) English translation: “for all values of x, P(x) is English translation: “for all values of x, P(x) is

true”true” English translation: “for all values of x, x+1>x English translation: “for all values of x, x+1>x

is true”is true”

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But is that always true?But is that always true? x P(x)x P(x)

Let x = the character ‘a’Let x = the character ‘a’ Is ‘a’+1 > ‘a’?Is ‘a’+1 > ‘a’?

Let x = the state of VirginiaLet x = the state of Virginia Is Virginia+1 > Virginia?Is Virginia+1 > Virginia?

You need to specify your universe!You need to specify your universe! What values x can representWhat values x can represent Called the “domain” or “universe of discourse” Called the “domain” or “universe of discourse”

by the textbookby the textbook

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Let the universe be the real numbers.Let the universe be the real numbers. Then, Then, x P(x) is truex P(x) is true

Let P(x) = x/2 < xLet P(x) = x/2 < x Not true for the negative numbers!Not true for the negative numbers! Thus, Thus, x P(x) is falsex P(x) is false

When the domain is all the real numbersWhen the domain is all the real numbers

In order to prove that a universal quantification is In order to prove that a universal quantification is true, it must be shown for ALL casestrue, it must be shown for ALL casesIn order to prove that a universal quantification is In order to prove that a universal quantification is false, it must be shown to be false for only ONE false, it must be shown to be false for only ONE casecase

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Universal quantification 4Universal quantification 4

Given some propositional function P(x)Given some propositional function P(x)

And values in the universe xAnd values in the universe x11 .. x .. xnn

The universal quantification The universal quantification x P(x) x P(x) implies:implies:

P(xP(x11) ) P(x P(x22) ) … … P(x P(xnn))

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Universal quantification 5Universal quantification 5

Think of Think of as a for loop: as a for loop:

x P(x), where 1 x P(x), where 1 ≤≤ x x ≤≤ 10 10

… … can be translated as …can be translated as …

for ( x = 1; x <= 10; x++ )for ( x = 1; x <= 10; x++ )is P(x) true?is P(x) true?

If P(x) is true for all parts of the for loop, then If P(x) is true for all parts of the for loop, then x P(x)x P(x) Consequently, if P(x) is false for any one value of the for loop, Consequently, if P(x) is false for any one value of the for loop,

then then x P(x) is falsex P(x) is false

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Quick surveyQuick survey

I understand universal quantificationI understand universal quantification

a)a) Very wellVery well

b)b) With some review, I’ll be goodWith some review, I’ll be good

c)c) Not reallyNot really

d)d) Not at allNot at all

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Chapter 2: Computer bugsChapter 2: Computer bugs

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Existential quantification 1Existential quantification 1

Represented by an bacwards E: Represented by an bacwards E: It means “there exists”It means “there exists” Let P(x) = x+1 > xLet P(x) = x+1 > x

We can state the following:We can state the following: x P(x)x P(x) English translation: “there exists (a value of) x English translation: “there exists (a value of) x

such that P(x) is true”such that P(x) is true” English translation: “for at least one value of English translation: “for at least one value of

x, x+1>x is true”x, x+1>x is true”

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Note that you still have to specify your Note that you still have to specify your universeuniverse If the universe we are talking about is all the If the universe we are talking about is all the

states in the US, then states in the US, then x P(x) is not truex P(x) is not true

Let P(x) = x+1 < xLet P(x) = x+1 < x There is no numerical value x for which x+1<xThere is no numerical value x for which x+1<x Thus, Thus, x P(x) is falsex P(x) is false

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Let P(x) = x+1 > xLet P(x) = x+1 > x There is a numerical value for which x+1>xThere is a numerical value for which x+1>x

In fact, it’s true for all of the values of x!In fact, it’s true for all of the values of x! Thus, Thus, x P(x) is truex P(x) is true

In order to show an existential quantification is In order to show an existential quantification is true, you only have to find ONE valuetrue, you only have to find ONE value

In order to show an existential quantification is In order to show an existential quantification is false, you have to show it’s false for ALL valuesfalse, you have to show it’s false for ALL values

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Given some propositional function P(x)Given some propositional function P(x)

And values in the universe xAnd values in the universe x11 .. x .. xnn

The existential quantification The existential quantification x P(x) x P(x) implies:implies:

P(xP(x11) ) P(x P(x22) ) … … P(x P(xnn))

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I understand existential I understand existential quantificationquantification





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A note on quantifiersA note on quantifiers

Recall that P(x) is a propositional functionRecall that P(x) is a propositional function Let P(x) be “x == 0”Let P(x) be “x == 0”

Recall that a proposition is a statement that is Recall that a proposition is a statement that is either true or falseeither true or false P(x) is not a propositionP(x) is not a proposition

There are two ways to make a propositional There are two ways to make a propositional function into a proposition:function into a proposition: Supply it with a valueSupply it with a value

For example, P(5) is false, P(0) is trueFor example, P(5) is false, P(0) is true Provide a quantifiactionProvide a quantifiaction

For example, For example, x P(x) is false and x P(x) is false and x P(x) is truex P(x) is true Let the universe of discourse be the real numbersLet the universe of discourse be the real numbers

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Binding variablesBinding variables

Let P(x,y) be x > yLet P(x,y) be x > y

Consider: Consider: x P(x,y)x P(x,y) This is not a proposition!This is not a proposition! What is y?What is y?

If it’s 5, then If it’s 5, then x P(x,y) is falsex P(x,y) is false

If it’s x-1, then If it’s x-1, then x P(x,y) is truex P(x,y) is true

Note that y is not “bound” by a quantifierNote that y is not “bound” by a quantifier

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Binding variables 2Binding variables 2

((x P(x)) x P(x)) Q(x) Q(x) The x in Q(x) is not bound; thus not a propositionThe x in Q(x) is not bound; thus not a proposition

((x P(x)) x P(x)) ( (x Q(x))x Q(x)) Both x values are bound; thus it is a propositionBoth x values are bound; thus it is a proposition

((x P(x) x P(x) Q(x)) Q(x)) ( (y R(y))y R(y)) All variables are bound; thus it is a propositionAll variables are bound; thus it is a proposition

((x P(x) x P(x) Q(y)) Q(y)) ( (y R(y))y R(y)) The y in Q(y) is not bound; this not a propositionThe y in Q(y) is not bound; this not a proposition

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Negating quantificationsNegating quantifications

Consider the statement:Consider the statement: All students in this class have red hairAll students in this class have red hair

What is required to show the statement is false?What is required to show the statement is false? There exists a student in this class that does NOT There exists a student in this class that does NOT

have red hairhave red hair

To negate a universal quantification:To negate a universal quantification: You negate the propositional functionYou negate the propositional function AND you change to an existential quantificationAND you change to an existential quantification ¬¬x P(x) = x P(x) = x ¬P(x)x ¬P(x)

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Negating quantifications 2Negating quantifications 2

Consider the statement:Consider the statement: There is a student in this class with red hairThere is a student in this class with red hair

What is required to show the statement is What is required to show the statement is false?false? All students in this class do not have red hairAll students in this class do not have red hair

Thus, to negate an existential quantification:Thus, to negate an existential quantification: Tou negate the propositional functionTou negate the propositional function AND you change to a universal quantificationAND you change to a universal quantification ¬¬x P(x) = x P(x) = x ¬P(x)x ¬P(x)

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I understand about bound variables I understand about bound variables and negating quantifiersand negating quantifiers





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Translating from EnglishTranslating from English

This is example 17, page 36This is example 17, page 36Consider “For every student in this class, Consider “For every student in this class, that student has studied calculus”that student has studied calculus”Rephrased: “For every student x in this Rephrased: “For every student x in this class, x has studied calculus”class, x has studied calculus” Let C(x) be “x has studied calculus”Let C(x) be “x has studied calculus” Let S(x) be “x is a student”Let S(x) be “x is a student”

x C(x)x C(x) True if the universe of discourse is all True if the universe of discourse is all

students in this classstudents in this class

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Translating from English 2Translating from English 2

What about if the unvierse of discourse is What about if the unvierse of discourse is all students (or all people?)all students (or all people?) x (S(x)x (S(x)C(x))C(x))

This is wrong! Why?This is wrong! Why? x (S(x)x (S(x)→→C(x))C(x))

Another option:Another option: Let Q(x,y) be “x has stuided y”Let Q(x,y) be “x has stuided y” x (S(x)x (S(x)→Q→Q(x, calculus))(x, calculus))

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This is example 18, page 36This is example 18, page 36

Consider:Consider: ““Some students have visited Mexico”Some students have visited Mexico” ““Every student in this class has visited Every student in this class has visited

Canada or Mexico”Canada or Mexico”

Let:Let: S(x) be “x is a student in this class”S(x) be “x is a student in this class” M(x) be “x has visited Mexico”M(x) be “x has visited Mexico” C(x) be “x has visited Canada”C(x) be “x has visited Canada”

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Consider: “Some students have visited Mexico”Consider: “Some students have visited Mexico” Rephrasing: “There exists a student who has visited Rephrasing: “There exists a student who has visited

Mexico”Mexico”

x M(x)x M(x) True if the universe of discourse is all studentsTrue if the universe of discourse is all students

What about if the unvierse of discourse is all What about if the unvierse of discourse is all people?people? x (S(x) x (S(x) →→ M(x)) M(x))

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

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Consider: “Every student in this class has Consider: “Every student in this class has visited Canada or Mexico”visited Canada or Mexico”

x (M(x)x (M(x)C(x)C(x) When the universe of discourse is all studentsWhen the universe of discourse is all students

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

Why isn’t Why isn’t x (S(x)x (S(x)(M(x)(M(x)C(x))) correct?C(x))) correct?

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Note that it would be easier to define Note that it would be easier to define V(x, y) as “x has visited y”V(x, y) as “x has visited y” x (S(x) x (S(x) V V(x,Mexico))(x,Mexico)) x (S(x)x (S(x)→→(V(x,Mexico) (V(x,Mexico) V(x,Canada)) V(x,Canada))

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Translate the statements:Translate the statements: ““All hummingbirds are richly colored”All hummingbirds are richly colored” ““No large birds live on honey”No large birds live on honey” ““Birds that do not live on honey are dull in color”Birds that do not live on honey are dull in color” ““Hummingbirds are small”Hummingbirds are small”

Assign our propositional functionsAssign our propositional functions Let P(x) be “x is a hummingbird”Let P(x) be “x is a hummingbird” Let Q(x) be “x is large”Let Q(x) be “x is large” Let R(x) be “x lives on honey”Let R(x) be “x lives on honey” Let S(x) be “x is richly colored”Let S(x) be “x is richly colored”

Let our universe of discourse be all birdsLet our universe of discourse be all birds

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Our propositional functionsOur propositional functions Let P(x) be “x is a hummingbird”Let P(x) be “x is a hummingbird” Let Q(x) be “x is large”Let Q(x) be “x is large” Let R(x) be “x lives on honey”Let R(x) be “x lives on honey” Let S(x) be “x is richly colored”Let S(x) be “x is richly colored”

Translate the statements:Translate the statements: ““All hummingbirds are richly colored”All hummingbirds are richly colored”

x (P(x)x (P(x)→S(x))→S(x)) ““No large birds live on honey”No large birds live on honey”

¬¬x (Q(x) x (Q(x) R(x)) R(x))Alternatively: Alternatively: x (x (¬Q(x) ¬Q(x) ¬R(x)) ¬R(x))

““Birds that do not live on honey are dull in color”Birds that do not live on honey are dull in color”x (x (¬R(x) → ¬S(x))¬R(x) → ¬S(x))

““Hummingbirds are small”Hummingbirds are small”x (P(x) x (P(x) → ¬Q(x))→ ¬Q(x))

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I understand how to translate I understand how to translate between English and quantified between English and quantified statementsstatements





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Today’s demotivatorsToday’s demotivators

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PrologProlog

A programming language using logic!A programming language using logic!Entering facts:Entering facts:

instructor (bloomfield, cs202)instructor (bloomfield, cs202)enrolled (alice, cs202)enrolled (alice, cs202)enrolled (bob, cs202)enrolled (bob, cs202)enrolled (claire, cs202)enrolled (claire, cs202)

Entering predicates:Entering predicates:teaches (P,S) :- instructor (P,C), enrolled (S,C)teaches (P,S) :- instructor (P,C), enrolled (S,C)

Extracting dataExtracting data?enrolled (alice, cs202)?enrolled (alice, cs202)

Result:Result:yesyes

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Prolog 2Prolog 2

Extracting dataExtracting data?enrolled (X, cs202)?enrolled (X, cs202)

Result:Result:alicealice

bobbob

claireclaire

Extracting dataExtracting data?teaches (X, alice)?teaches (X, alice)

Result:Result:bloomfieldbloomfield

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I felt I understood the material in this I felt I understood the material in this slide set…slide set…





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The pace of the lecture for this The pace of the lecture for this slide set was…slide set was…

a)a) FastFast

b)b) About rightAbout right

c)c) A little slowA little slow

d)d) Too slowToo slow

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How interesting was the material in How interesting was the material in this slide set? Be honest!this slide set? Be honest!

a)a) Wow! That was SOOOOOO cool!Wow! That was SOOOOOO cool!

b)b) Somewhat interestingSomewhat interesting

c)c) Rather bortingRather borting

d)d) ZzzzzzzzzzzZzzzzzzzzzz

Documents

1 Predicates and Quantifiers CS/APMA 202, Spring 2005 Rosen, section 1.3 Aaron Bloomfield