Negation of Quantifiers

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Discrete Structures

Predicate Logic 2

Dr. Muhammad Humayoun

Assistant Professor

COMSATS Institute of Computer Science, Lahore.

mhumayoun@ciitlahore.edu.pk

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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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Exercise

B(x): “x is a baby”

ignorant(x): “x is ignorant”

vain(x): “x is vain”

Universe: The set of all people.

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Exercise

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Exercise

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Exercise

professor(x): “x is a professor”

• _{No professors are ignorant.}

• _{It is not the case that there exists an x such that x}

is a professor and x is ignorant.

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Exercise

• _{[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.]

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Exercise

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Exercise

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Exercise

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Exercise

professor(x): “x is a professor”

ignorant(x): “x is ignorant”

vain(x): “x is vain”

• _{All ignorant people are vain.}

• _{For all people x, if x is ignorant then x is vain.}

• _{It is logically equivalent to}

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Exercise

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Exercise

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Exercise

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Exercise

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Exercise

• _{No professors are vain}

• _{It is not the case that there is an x such that x is professor} and x is vain.

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Exercise

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Exercise

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Precedence of Quantifiers

• _{The quantifiers and have higher}

precedence then all logical operators from

propositional calculus.

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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:

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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:

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Meanings of multiple quantifiers

Suppose = “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}

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Meanings of multiple quantifiers

•

– _{For every value of x we can find a (possibly}

different) y so that P(x,y) is true.

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Meanings of multiple quantifiers

•

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

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Meanings of multiple quantifiers

•

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

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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

• _{There is a real number z such that for all real}

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Example

• _True

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Example

• _True

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

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Example

• _True

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

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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}

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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}

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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“}

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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“}

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• _{The sum of two positive integers is always}

positive.

• _{What is domain above?} • _Integers

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positive.

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positive.

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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”}

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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}

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Bound and free variables

A variable is bound if it is known or quantified.

Otherwise, it is free.

Examples:

_P(_x₎ _x_{is free}

P(5) x is bound to 5

_x_P(_x₎ _x_{is bound by quantifier}

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