Post on 22-Feb-2016
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Discrete StructuresPredicate Logic 2
Dr. Muhammad HumayounAssistant Professor
COMSATS Institute of Computer Science, Lahore.mhumayoun@ciitlahore.edu.pk
https://sites.google.com/a/ciitlahore.edu.pk/dstruct/
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Negation of Quantifiers
• ???
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Negation of Quantifiers
• ???
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Negation of Quantifiers
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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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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. (Ambiguous)• All/Some babies are ignorant
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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. (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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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.• [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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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.• [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.]
professors 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.• [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.]
• All professors are not 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.
• All (and all of them) professors are not 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.
• 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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Exerciseprofessor(x): “x is a professor” ignorant(x): “x is ignorant” vain(x): “x is vain” Universe: The set of all people.
• 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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Exerciseprofessor(x): “x is a professor” ignorant(x): “x is ignorant” vain(x): “x is vain” Universe: The set of all people.
• 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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Exerciseprofessor(x): “x is a professor” ignorant(x): “x is ignorant” vain(x): “x is vain” Universe: The set of all people.
• 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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Exerciseprofessor(x): “x is a professor” ignorant(x): “x is ignorant” vain(x): “x is vain” Universe: The set of all people.
• 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.
Useful
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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 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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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 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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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 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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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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Quantifiers with Restricted Domain
• –
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Quantifiers with Restricted Domain
• –
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Quantifiers with Restricted Domain
• –
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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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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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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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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
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
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
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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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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• The sum of two positive integers is always positive.
• What is domain above?• Integers• If domain is “+ve integers”
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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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• 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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• 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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• 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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• 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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• 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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• 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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• There is a woman who has taken a flight on every airline in the world.
• 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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•
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• There is a woman who has taken a flight on every airline in the world.
• 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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• There is a woman who has taken a flight on every airline in the world.
• 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