discrete structures predicate logic 2

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Discrete StructuresPredicate Logic 2

Dr. Muhammad HumayounAssistant Professor

COMSATS Institute of Computer Science, [email protected]

https://sites.google.com/a/ciitlahore.edu.pk/dstruct/

mailto:[email protected]

https://sites.google.com/a/ciitlahore.edu.pk/dstruct/

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Negation of Quantifiers

• ???

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• ???

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• ???

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ExerciseB(x): “x is a baby” ignorant(x): “x is ignorant” vain(x): “x is vain” Universe: The set of all people.

• Babies are ignorant.

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• Babies are ignorant. (Ambiguous)• All/Some babies are ignorant

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• Babies are ignorant. (Ambiguous)• All babies are ignorant

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Exerciseprofessor(x): “x is a professor” ignorant(x): “x is ignorant” vain(x): “x is vain” Universe: The set of all people.

• No professors are ignorant.• It is not the case that there exists an x such that x

is a professor and x is ignorant.

• It is not the case that all professors are ignorant.

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• No professors are ignorant.• [There is no such professor who is ignorant]• [It is not the case that there is an x such that x is a

professor and x is ignorant.]

• It is not the case that all professors are ignorant.

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professors are ignorant.

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• All professors are not ignorant

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• No professors are ignorant.

• All (and all of them) professors are not ignorant.

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• All ignorant people are vain. • For all people x, if x is ignorant then x is vain.

• It is logically equivalent to

• There is no such person x such that he is ignorant and not vain.

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Useful

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• No professors are vain• It is not the case that there is an x such that x is professor

and x is vain.

• For all people x, if x is a professor then x not vain.

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and x is vain.


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and x is vain.


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Precedence of Quantifiers

• The quantifiers and have higher precedence then all logical operators from propositional calculus.

• e.g. is the disjunction of .

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Quantifiers with Restricted Domain

• –

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• –

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• –

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• –

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Nested Quantifiers

“For all , there exists a such that”. Example: where and are integers

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Nested Quantifiers

“For all , there exists a such that”. Example: where and are integers

There exists an x such that for all , is true” Example:

• THINK QUANTIFICATION AS LOOPS

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Meanings of multiple quantifiersSuppose = “x likes y.”Domain of x: {St1, St2}; Domain of y: {DS, Calculus}

• – true for all x, y pairs.

– true for at least one x, y pair.

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

– There is at least one x for which P(x,y) is always true.

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• Quantification order is not commutative

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Example

Domain: Real numbers

• True/False??? • For all real numbers x and for all real numbers y

there is a real number z such that .• True • True/False???• There is a real number z such that for all real

numbers x and for all real numbers y it is true that .• False

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Example





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Example





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Example





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From Nested Quantifiers to English

• F (a, b): “a and b are friends” • Domain: All students in COMSATS.

• There is a student x such that for all students y and all students z other than y, if x and y are friends and x and z are friends, then y and z are not friends.

• There is a student none of whose friends are also friends with each other.

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From Nested Quantifiers to English

• F (a, b): “a and b are friends” • Domain: All students in COMSATS.

• There is a student x such that for all students y and all students z other than y, if x and y are friends and x and z are friends, then y and z are not friends.

• There is a student none of whose friends are also friends with each other.

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From English to Nested Quantifiers

• "If a person is female and is a parent, then this person is someone's mother“

• For every person x , if person x is female and person x is a parent, then there exists a person y such that person x is the mother of person y.“– F(x): “x is female”– P(x): “x is a parent“– M(x, y) : “x is the mother of y”

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From English to Nested Quantifiers

• "If a person is female and is a parent, then this person is someone's mother“

• For every person x , if person x is female and person x is a parent, then there exists a person y such that person x is the mother of person y.“– F(x): “x is female”– P(x): “x is a parent“– M(x, y) : “x is the mother of y”

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• The sum of two positive integers is always positive.

• What is domain above?• Integers• If domain is “+ve integers”

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• Everyone has exactly one best friend• For every person x , person x has exactly one best

friend.

• B(x,y): “x has best friend y”

• Exactly one best friend ????

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friend.



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friend.



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friend.



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friend.



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friend.



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• There is a woman who has taken a flight on every airline in the world.

• Domains: people airlines flights• W(x): x is a woman• F(x, f): x has taken flight f• A(f, a): flight f belongs to airline a

•

•

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• Domains: woman airlines flights• P(w, f): Woman w has taken flight f• Q(f, a): flight f belongs to airline a

•

•

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• Domains: woman airlines flights• P(w, f): Woman w has taken flight f• Q(f, a): flight f belongs to airline a

•

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• Domains: woman airlines flights• R(w, f, a): Woman w has taken flight f on airline a

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Bound and free variablesA variable is bound if it is known or quantified.

Otherwise, it is free.

Examples:P(x) x is freeP(5) x is bound to 5x P(x) x is bound by quantifier

Reminder: in a proposition, all variables must be bound.

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Negating Nested Quantifiers

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Do Exercises

discrete structures predicate logic 2

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