All dogs sleep.
Some four-legged animals are dogs.

∀x (Dog(x) → Sleep(x))
∃x ∃y (FourLegged(x) ∧ Animal(x) ∧ Dog(x) ∧ FourLegged(y) ∧ Animal(y) ∧ Dog(y) ∧ ¬(x=y))

∃x ∃y (FourLegged(x) ∧ Animal(x) ∧ Sleeps(x) ∧ FourLegged(y) ∧ Animal(y) ∧ Sleeps(y) ∧ ¬(x=y))


Everyone who is entitled to national social insurance coverage can have their medical bills partially covered.
All PRC nationals are entitled to national social insurance coverage.
Everyone in the Franco-China diplomatic conference is either a PRC national or a French national, but not both.
All French nationals are citizens of the European Union.
All Spanish nationals are citizens of the European Union.
No North Korean nationals are citizens of the European Union.
Mei is at the Franco-China diplomatic conference.
Either Mei is a North Korean and can have medical bills partially covered, or neither is true.

∀x (EntitledTo(x, nationalSocialInsuranceCoverage) → CanHavePartiallyCovered(x, medicalBills))
∀x (PRCNational(x) → EntitledTo(x, nationalSocialInsuranceCoverage))
∀x (In(x, franco-ChinaDiplomaticConference) → PRCNational(x) ⊕ FrenchNational(x))
∀x (FrenchNational(x) → CitizenOf(x, europeanUnion))
∀x (SpanishNational(x) → CitizenOf(x, europeanUnion))
∀x (NorthKoreanNational(x) → ¬CitizenOf(x, europeanUnion))
In(mei, franco-ChinaDiplomaticConference)
¬(NorthKoreanNational(mei) ⊕ CanHavePartiallyCovered(mei, medicalBills))


If Mei is either a North Korean or a Spanish national, then Mei is either both a French national and a citizen of the European Union, or neither a French national nor a citizen of the European Union.

NorthKoreanNational(mei) ⊕ SpanishNational(mei) → ¬(FrenchNational(mei) ⊕ European(mei))


No people who do not admit a mistake are good teachers.
Some well-informed people are people who do not admit a mistake.

∀x (¬Admit(x, mistake) → ¬GoodTeacher(x))
∃x ∃y (WellInformed(x) ∧ WellInformed(y) ∧ ¬Admit(x, mistake) ∧ ¬Admit(y, mistake) ∧ ¬(x=y))

Some good teachers are not well-informed people.

∃x ∃y (GoodTeacher(x) ∧ GoodTeacher(y) ∧ ¬WellInformed(x) ∧ ¬WellInformed(y) ∧ ¬(x=y))


Philatelic literature is divided into the following categories: Stamp catalogs, Periodicals, Auction catalogs, Books, Bibliographies, and Background Material.
Mort is not a Stamp catalog.
Mort is not a periodical, auction catalog, bibliography, or background material.
Mort is a piece of Philatelic literature.

∀x (PhilatelicLit(x) → (Stamp(x) ∨ Periodical(x) ∨ Auction(x) ∨ Book(x) ∨ Bibliography(x) ∨ Background(x)))
¬Stamp(mort)
¬(Periodical(mort) ∨ Auction(mort) ∨ Bibliography(mort) ∨ Background(mort))
PhilatelicLit(mort)


Eragon is a piece of Philatelic literature.


Adventures of Rusty is a drama film and children's film.
Columbia Pictures produced Adventures of Rusty.
Tintin was produced by Paramount.
Tintin is an adventure film.

DramaFilm(adventuresOfRusty) ∧ ChildrensFilm(adventuresOfRusty)
Produces(columbiaPictures, adventuresOfRusty)
Produces(paramount, tintin)
AdventureFilm(tintin)

Columbia pictures produced some drama film.

∃x (DramaFilm(x) ∧ Produces(columbiaPictures, x))


Columbia pictures produced some adventure film.

∃x (AdventureFilm(x) ∧ Produces(columbiaPictures, x))


∃x (ChildrensFilm(x) ∧ Produces(paramount, x))


∃x (AdventureFilm(x) ∧ Produces(paramount, x))


Deng Xiaoping served as the paramount leader of the People's Republic of China.
Deng Xiaoping was praised for his reaffirmation of the reform program, as well as the reversion of Hong Kong to Chinese control and the return of Macau.
As the party's Secretary-General under Mao and Vice Premier in the 1950s, Deng Xiaoping presided over the Anti-Rightist Campaign launched by Mao.
Deng Xiaoping became instrumental in China's economic reconstruction following the disastrous Great Leap Forward.
Mao Zedong died in 1976.
After Mao Zedong's death, Deng Xiaoping gradually rose to supreme power.

ParamountLeaderOf(dengXiaoping, peoplesRepublicOfChina)
PraisedFor(dengXiaoping, reaffirmationOfReformProgram) ∧ PraisedFor(dengXiaoping, reversionOfHongKong) ∧ PraisedFor(dengXiaoping, returnOfMacau)
PartysSecretaryGeneral(dengXiaoping) ∧ Under(dengXiaoping, mao) ∧ VicePremierInThe1950s(dengXiaoping) ∧ PresidedOver(dengXiaoping, antiRightistCampaign) ∧ LaunchedBy(antiRightistCampaign, mao)
InstrumentalIn(dengXiaoping, chinasEconomicReconstruction) ∧ Following(chinasEconomicReconstruction, greatLeapForward) ∧ Disastrous(greatLeapForward)
DiedIn(mao, year1976)
GraduallyRoseTo(dengXiaoping, supremePower)

The paramount leader of the PRC was also the vice premier.

∃x (ParamountLeaderOf(x, prc) ∧ VicePremier(x))


Deng Xiaoping presided over something launched by someone he was under.

∃x ∃y (PresidedOver(dengxiaoping, x) ∧ Under(dengxiaoping, y) ∧ LaunchedBy(x, y))


The person instrumental in china's economic reconstruction gradually rose to supreme power.

∃x (InstrumentalIn(x, chinaseconomicreconstruction) ∧ GraduallyRoseTo(x, supremepower))


All imaginative processes that Dan knows are results of creative processes.
All science fiction that Dan knows comes from an imaginative process.
Everthing that Dan knows comes from either science-fiction or realistic fiction.
No facts that Dan knows have proven to be false.
Dan knows that Dune is science fiction or has proven to be false.

∀x ((Knows(dan, x) ∧ ImaginativeProcess(x)) → ResultOf(x, creativeProcess))
∀x ((Knows(dan, x) ∧ ScienceFiction(x)) → ImaginativeProcess(x))
∀x (Knows(dan, x) → (ScienceFiction(x) ⊕ RealisticFiction(x)))
∀x ((Knows(dan, x) ∧ Fact(x)) → ¬ProvedToBe(x, false))
(Knows(dan, dune) ∧ ScienceFiction(dune)) ∨ ProvedToBe(dune, false))


Dune is a result of creative and imaginative process.

ResultOf(dune, creativeProcess) ∧ ImaginativeProcess(dune)


Dune is either a result of creative processes or came from an imaginative process.

ResultOf(dune, creativeProcess) ⊕ ImaginativeProcess(dune)


Dune is a result of creative processes and is science fiction.

ResultOf(dune, creativeProcess) ∧ ScienceFiction(dune))


Dune is either a result of creative processes or is science fiction.

Knows(dan, dune) ∧ (ResultOf(dune, creativeProcess)  ⊕ ScienceFiction(dune))


If Dune is a result of creative and imaginative processes, then Dune is not a result of creative processes and science-fiction.

(ResultOf(dune, creativeProcess)  ∧ ImaginativeProcess(dune)) → (¬ResultOf(dune, creativeProcess) ∧ ¬ScienceFiction(dune))


If Dune is either a fact and a result of creative processes, or neither a fact nor a result of creative processes, then Dune is a result of creative processes and science-fiction.

Knows(dan, dune) ∧ (¬(Fact(dune) ⊕ ResultOf(dune, creativeProcess))) → (ResultOf(dune, creativeProcess) ∧ ScienceFiction(dune))


If Dune is science-fiction, then Dune is not a result of creative processes and science-fiction.

Knows(dan, dune) ∧ (ScienceFiction(dune)) → (¬(ResultOf(dune, creativeProcess)  ∧ ScienceFiction(dune)))


If Dune is not a result of creative processes and science-fiction, then Dune neither came from an imaginative process nor proved to be false.

Knows(dan, dune) ∧ (¬(ResultOf(dune, creativeProcess)  ∧ ScienceFiction(dune))) → (¬(ImaginativeProcess(dune) ∨ ProvedToBe(dune, false)))


If Dune is did not come from imaginative process and is not science-fiction, then Dune is neither a result of creative processes nor came from an imaginative process.

Knows(dan, dune) ∧ (¬(ImaginativeProcess(dune) ∧ ScienceFiction(dune))) → (¬(ResultOf(dune, creativeProcess)  ∨ ImaginativeProcess(dune)))


American superheroes come from either the DC Universe or Marvel Universe.
Captain America is one of America's top-ten favorite superheroes
Captain America does not come from the DC Universe.
America's top-ten favorite superheroes speak English.
Some superheroes speak both English and Spanish.

∀x (Superhero(x) ∧ American(x) → ComeFrom(x, dCUniverse) ⊕ ComeFrom(x, marvelUniverse))
American(captainAmerica) ∧ TopTenFavorite(captainAmerica) ∧ Superhero(captainAmerica)
¬ComeFrom(captainAmerica, dCUniverse)
∀x (American(x) ∧ TopTenFavorite(x) ∧ Superhero(x)  → Speak(x, english))
∃x (Superhero(x) → (Speak(x, english) ∧ Speak(x, spanish)))


Captain America comes from the Marvel universe.


Robert Zimmer was a philosopher born in Germany.
Robert Zimmer is an essayist.
Robert Zimmer was born in 1953.
Every essayist is a writer.

BornIn(robertZimmer, germany) ∧ Philosopher(robertZimmer)
Essayist(robertZimmer)
BornIn(robertZimmer, yr1953)
∀x (Essayist(x) → Writer(x))


All people who repay their loans on time have a high credit score.
Some people with high credit scores and high salaries are approved for mortgages.
John has a high salary.

∀x (RepayOnTime(x) → Has(x, highCreditScore))
∃x ((Has(x, highCreditScore) ∧ Has(x, highSalary)) → ApprovedFor(x, mortgage))
Has(john, highSalary)

If John repays his loans on time, he will be approved for a mortgage.

RepayOnTime(john) → ApprovedFor(john, mortgage)


All students are members of the university.
All graduate students are students.
All PhD students are graduate students.
Some PhD students are Teaching Fellows.
If John is not a PhD student, then he is not a member of the university.
If John is a Teaching Fellow, then he is a PhD student or a graduate student.

∀x (Student(x) → MemberOf(x, university))
∀x (GraduateStudent(x) → Student(x))
∀x (PhDStudent(x) → GraduateStudent(x))
∃x (PhDStudent(x) ∧ TeachingFellow(x))
¬PhDStudent(john) → ¬MemberOf(john, university)
TeachingFellow(john) → PhDStudent(john) ⊕ GraduateStudent(john)


Belgium, France, and Germany are European countries.
Paris is the capital of France.
The Eiffel Tower is one of the main tourist attractions located in Paris.
Some people who live in Belgium speak French.
If John goes to Europe, he will see some tourist attractions.
John speaks French.

EuropeanCountry(belgium) ∧ EuropeanCountry(france) ∧ EuropeanCountry(germany)
CapitalOf(paris, france)
TouristAttraction(eiffelTower) ∧ LocatedIn(eiffelTower, paris)
∃x (LiveIn(x, belgium) → Speak(x, french))
∃x (GoTo(john, europe) → (See(john, x) ∧ TouristAttraction(x)))
Speak(john, french)

If John goes to Europe, he will see the Eiffel Tower.

GoTo(john, europe) → See(john, eiffelTower)


The Eiffel Tower is located in the capital of France.

∃x (CapitalOf(x, france) ∧ LocatedIn(eiffelTower, x))


All sports cars are loud.
No loud cars are electric.
If a car is a Ferrari, then it is a sports car.
All cars made in Maranello are Ferraris.
The Toyota Prius is made in Maranello or is a loud car, or both.

∀x (SportsCar(x) → LoudCar(x))
∀x (LoudCar(x) → ¬ElectricCar(x))
∀x (Ferrari(x) → SportsCar(x))
∀x ((Car(x) ∧ MadeIn(x, maranello)) → Ferrari(x))
(Car(toyotaPrius) ∧ MadeIn(toyotaPrius, maranello)) ∨ LoudCar(toyotaPrius)


The Toyota Prius is not an electric car.


The Toyota Prius is a equipped with a Ferrari V12 engine.


If The Toyota Prius is a Ferrari or a loud car, then The Toyota Prius is an electric car.

Ferrari(toyotaPrius) ∨ LoudCar(toyotaPrius) → ElectricCar(toyotaPrius)


If something is a plant, then it is not a cute animal. Simeng: All plants are not cute animals.
All flowers are plants.
Every kitten is a cute animal.
If something is grown in a garden, then it is a flower.
Piper is a kitten or a cute animal.

∀x (Plant(x) → ¬CuteAnimal(x))
∀x (Flower(x) → Plant(x))
∀x (Kitten(x) → CuteAnimal(x))
∀x (GrownIn(x, garden) → Flower(x))
Kitten(piper) ∨ CuteAnimal(piper)


Guam sent an athlete to the Calgary Winter Olympics.
If Guan sent an athlete to the Calgary Winter Olympics, then the athelete participated in the Olympics in 1988.
Judd Bankert is the only athlete from Guam who has ever competed in the Winter Olympics.

∃x (Send(guam, athlete, calgaryWinterOlympics))
∀x (Athlete(x) ∧ SendTo(guam, x, calgaryWinterOlympics) → ParticipatedIn(x, winterOlympics, year1988))
∀x ∀y (Athlete(x) ∧ From(x, guam) ∧ ParticipatedIn(x, winterOlympics, y) → x=juddBankert)

Judd Bankert competed in the 1988 Winter Olympics.


Guam has participated in the Summer Olympics at least once.

∃x (ParticipatedIn(guam, summerOlympics, x))


Michael O'Donnell is a British physician, journalist, author, and broadcaster.
One of the word-setters of My Word! was Michael O'Donnell.
The magazine World Medicine was edited by Michael O'Donnell.
Michael O'Donnell was born in Yorkshire as the son of a general practitioner.

British(michael) ∧ Physician(michael) ∧ Journalist(michael) ∧ Author(michael) ∧ Broadcaster(michael)
WordSetter(michael)
Magazine(worldMedicine) ∧ EditedBy(worldMedicine, michael)
BornIn(michael, yorkshire) ∧ ∃x(SonOf(michael, x) ∧ GeneralPractitioner(x))

The son of a general practitioner was a word-setter of My Word!.

∃x ∃y (SonOf(x, y) ∧ GeneralPractitioner(y) ∧ WordSetter(x))


There are no journalists that were born in Yorkshire.

∀x (Journalist(x) → ¬BornIn(x, yorkshire))


There is a son of a general practitioner that is not an author.

∃x ∃y (Son(x, y) ∧ GeneralPractitioner(y) ∧ ¬Author(x))


No homework is fun.
Some reading is homework.

∀x (Homework(x) → ¬Fun(x))
∃x (Reading(x) ∧ Homework(x))


The handbrake of a car is either up or down.
The handbrake is down when a car is parked.

∀x ∀y (HandbrakeOf(x, y) ∧ Car(y) → Up(x) ⊕ Down(x))
∀x ∀y (HandbrakeOf(x, y) ∧ Parked(y) ∧ Car(y) → Down(x))

The handbrake is up when some cars are parked.

∃x ∃y (HandbrakeOf(x, y) ∧ Parked(y) ∧ Car(y) ∧ Up(x))


All people in this midwest town who own horse ranches regularly ride horses for pleasure and sport.
All people in this midwest town with a lot of disposable income have a horse ranch.
If people in this midwest town compete in horse dressage shows, then they have a lot of disposable income.
If people in this midwest town compete in horse dressage shows, then they have invested in high-quality equestrian gear and equipment.
If people in this midwest town regularly ride horses for pleasure and sport, then they do not live in cramped residential buildings.
Manny is in this midwest town, and she either has a horse ranch and lives in a cramped residential building, or she does neither.

∀x (InThisMidwestTown(x) ∧ Have(x, horseRanch) → RegularlyRideHorseForPleasure(x))
∀x (InThisMidwestTown(x) ∧ Have(x, disposableIncome) → Have(x, horseRank))
∀x (InThisMidwestTown(x) ∧ CompeteIn(x, horseDressageShow) → Have(x, disposableIncome))
∀x (InThisMidwestTown(x) ∧ CompeteIn(x, horseDressageShow) → InvestedIn(x, equestrianGearAndEquipment))
∀x (InThisMidwestTown(x) ∧ RegularlyRideHorseForPleasure(x) → ¬LiveIn(x, crampedBuilding))
InThisMidwestTown(manny) ∧ ¬(Have(manny, horseRanch) ⊕ LiveIn(manny, crampedBuilding))

Manny regularly rides horses for pleasure and sport.


Manny competes in horse dressage shows and has invested in high-quality equestrian equipment and gear.

CompeteIn(manny, horseDressageShow) ∧ InvestedIn(manny, equestrianGearAndEquipment)


If Manny either has a horse ranch or competes in horse dressage shows, then Manny has not invested in high-quality equestrian equipment and gear.

¬(HaveAHorseRanch(manny) ⊕ CompeteIn(manny, horseDressageShow)) → ¬InvestedIn(manny, equestrianGearAndEquipment)


A roundel is a rounded artillery fortification.
A roundel is not higher than adjacent walls.
Cannons can be deployed on artillery fortifications.
Roundels are the oldest artillery fortifications.
Battery towers are artillery fortifications.

∀x (Roundel(x) → (Rounded(x) ∧ ArtilleryFortification(x)))
∀x ∀y ((Roundel(x) ∧ AdjacentWalls(x,y)) → ¬Higher(x, y))
∀x (ArtilleryFortification(x) → DeployCannons(x))
∀x ∀y ((Roundel(x) ∧ ArtilleryFortification(y)) → Older(x, y))
∀x (BatteryTower(x) → ArtilleryFortification(x))

Cannons can be deployed on battery towers.


∀x ∀y ((Roundel(x) ∧ BatteryTower(y)) → Older(x, y))


Battery towers are higher than adjacent walls.

∀x ∀y ((BatteryTower(x) ∧ AdjacentWall(x,y)) → Higher(x, y))


Tissues are soft.
Some papers are tissues.

∀x (Tissue(x) → Soft(x))
∃x ∃y (Paper(x) ∧ Paper(x) ∧ Tissue(x) ∧ Tissue(y) ∧ ¬(x=y))

∃x ∃y (Paper(x) ∧ Paper(y) ∧ Hard(x) ∧ Hard(y) ∧ ¬(x=y))


All volunteers receive intangible benefits for their work.
Volunteers work regularly or on an as-needed basis.
Some volunteers are trained.
Volunteers work in groups or individually.
Environmental volunteers contribute toward environmental management or conservation.
Participating in natural disaster response is an example of volunteers working in groups on an as-needed basis.

∀x (Volunteer(x) → Receive(x, intangibleBenefit))
∀x (Volunteer(x) → WorkRegularly(x) ⊕ WorkAsNeeded(x))
∃x (Volunteer(x) → Trained(x))
∀x (Volunteer(x) → (WorkInGroup(x) ∨ WorkIndividually(x)))
∀x (Volunteer(x) ∧ Environmental(x) → (ContributeTo(x, environmentalManagement) ∨ ContributeTo(x, environmentalConservation)))
∃x (Volunteer(x) ∧ ContributeTo(x, naturalDisasterResponse) → WorkInGroup(x) ∧ WorkAsNeeded(x))

Volunteers who participate in natural disaster response receive intangible benefits for their work.

∀x (Volunteer(x) ∧ ContributeTo(x, naturalDisasterResponse) → Receive(x, intangibleBenefit))


Environmental volunteers work in groups.

∀x (Volunteer(x) ∧ Environmental(x) → WorkInGroup(x))


All people in this tech company who are consistent and enjoy sticking to their regular routines do not like surprises.
People in this tech company who wear the same flannel shirts every day are consistent and enjoy sticking to their regular routines.
People in this tech company who do not like shopping for clothes wear the same flannel shirts every day.
Old people living in stable homes do not like surprises.
People in this tech company who have very high energy and are impulsive like surprises.
Mike works in this tech company.
If Mike is not a person who wears the same flannel shirts every day, has very high energy, and is impulsive, then Mike either is very consistent and enjoys sticking to his regular routines or does not like surprises.

∀x (InThisTechCompany(x) ∧ Consistent(x) ∧ StickTo(x, theirRegularRoutine) → ¬Like(x, surprise))
∀x (InThisTechCompany(x) ∧ ∃y (flannelShirt(y) ∧ WearEveryday(x, y)) → Consistent(x) ∧ StickTo(x, theirRegularRoutine))
∀x (InThisTechCompany(x) ∧ ¬LikeShoppingFor(x, clothes) → ∃y (flannelShirt(y) ∧ WearEveryday(x, y)))
∀x (InThisTechCompany(x) ∧ Old(x) ∧ LiveIn(x, stableHome) → ¬Like(x, surprise))
∀x (InThisTechCompany(x) ∧ Have(x, highEnergy) ∧ Impulsive(x) → ¬Like(x, surprise))
InThisTechCompany(mike)
¬(∃y (flannelShirt(y) ∧ WearEveryday(x, y)) ∧ Have(mike, highEnergy) ∧ Impulsive(mike)) → (Consistent(mike) ∧ StickTo(mike, theirRegularRoutine)) ⊕ ¬Like(mike, surprise)

Mike is an old person living in a stable home.


If Mike wears the same flannel shirts every day or does not like shopping for clothes, then Mike is neither an old person living in a stable home nor does he like shopping for clothes.

(∃y (flannelShirt(y) ∧ WearEveryday(mike, y)) ∨ ¬LikeShoppingFor(mike, clothes)) → ¬(Old(mike) ∧ LiveIn(mike, stableHome)) ∧ ¬LikeShoppingFor(mike, clothes)


If Mike is not an old person living in a stable home and does not like shopping for clothes, then Mike does not like shopping for clothes.

¬(Old(mike) ∧ LiveIn(mike, stableHome)) ∧ ¬LikeShoppingFor(mike, clothes)) → ¬LikeShoppingFor(mike, clothes)


Adam owns cars.
Adam has a favorite car.
Among the cars he owns, Adam's favorite car is European.
Adam broke his favorite car.

∃x∃y (Car(x) ∧ Car(y) ∧ (x≠y) ∧ Owns(adam, x))
∃x (Car(x) ∧ Favorite(adam, x))
∀x ((Car(x) ∧ Owns(adam, x) ∧ Favorite(adam, x)) → European(x))
∀x ((Car(x) ∧ Owns(adam, x) ∧ Favorite(adam, x)) → Broke(adam, x))


There are no buildings in New Haven higher than 400 meters.
All buildings managed by Yale Housing are in New Haven.
All Manhattan skyscrapers are higher than 400 meters.
All buildings owned by Bloomberg are in Manhattan.
All buildings with the Bloomberg logo are buildings owned by Bloomberg.
Tower A is neither a building in New Haven nor a skyscraper in Manhattan.
Tower B is a skyscraper building in Manhattan with a Bloomberg logo.

∀x ((Buildings(x) ∧ In(x, newHaven)) → ¬HigherThan(x, num400))
∀x ((Buildings(x) ∧ ManagedBy(x, yaleHousing)) → In(x, newHaven))
∀x ((Buildings(x) ∧ Skyscraper(x) ∧ In(x, manhattan)) → HigherThan(x, num400))
∀x ((Buildings(x) ∧ OwnedBy(x, bloomberg)) → Skyscraper(x) ∧ In(x, manhattan))
∀x ((Buildings(x) ∧ HasLogo(x, bloomberg)) → OwnedBy(x, bloomberg))
Buildings(towerA) ∧ (¬InNewHaven(towerA)) ∧ (¬ManhattanSkyscraper(towerA))
Buildings(towerB) ∧ HasLogo(towerB, bloomberg) ∧ Skyscraper(towerB) ∧ In(towerB, manhattan)


Tower A is a building with the Bloomberg logo or it is managed by Yale Housing.

HasLogo(towerB, bloomberg) ∨ ManagedBy(x, yaleHousing)


Tower A is neither a building with the Bloomberg logo nor managed by Yale Housing.

¬HasLogo(towerB, bloomberg) ∧ (¬ManagedBy(x, yaleHousing))


No fish are birds.
An osprey is a bird.
A carp is a fish.
All goldfish are carp.
If Bubbles is either an osprey or a goldfish, then Bubbles is not also a fish.

∀x (Fish(x) → ¬Bird(x))
∀x (Osprey(x) → Bird(x))
∀x (Carp(x) → Fish(x))
∀x (Goldfish(x) → Carp(x))
Osprey(bubbles) ⊕ Goldfish(bubbles) → ¬Fish(bubbles)


Mr. and Mrs. Smith make a travel plan: they want to go to a city in California or Florida where neither of them has ever been.
The cities in California that they are interested in are San Francisco, Los Angeles, and San Diego.
Cities in Florida that they are interested in are Orlando and Miami.
Mr. Smith has been to two cities in California.
Mrs. Smith has been to one city in Florida.

∀x (WantToGoTo(mr.AndMrs.Smith, x) ∧ City(x) → (California(x) ∨ Florida(x)) ∧ NeverGo(x))
City(sanFrancisco) ∧ California(sanFrancisco) ∧ WantToGoTo(mr.AndMrs.Smith, sanFrancisco) ∧ City(losAngeles) ∧ California(losAngeles) ∧ WantToGoTo(mr.AndMrs.Smith, losAngeles) ∧ City(sanDiego) ∧ California(sanDiego) ∧ WantToGoTo(mr.AndMrs.Smith, sanDiego)
City(orlando) ∧ Florida(orlando) ∧ WantToGo(mr.AndMrs.Smith, orlando) ∧ City(miami) ∧ Florida(miami) ∧ WantToGo(mr.AndMrs.Smith, miami)
∃x ∃y ∀z (¬(x=z) ∧ ¬(y=z) ∧ ¬(x=y) ∧ City(x) ∧ City(y) ∧ City(z) ∧ California(x) ∧ California(y) ∧ California(z) → Visit(mr.smith, x) ∧ Visit(mr.smith, y) ∧ ¬Visit(mr.smith, z))
∃x ∀y (¬(x=y) ∧ City(x) ∧ City(y) ∧ Florida(x) ∧ Florida(y) → Visit(mrs.smith, x) ∧ ¬Visit(mrs.smith, y))

∃x (City(x) ∧ Visit(mr.smith, sanFrancisco))


They have at leat one candidate city in Florida to visit.

∃x (WantToGoTo(x) ∧ City(x) ∧ Florida(x))


They have at least two candidate cities in California to visit.

∃x ∃y (¬(x=y) ∧ City(x) ∧ City(y) ∧ WantToGoTo(mr.AndMrs.Smith, x) ∧ California(x) ∧ WantToGoTo(mr.AndMrs.Smith, y) ∧ California(y))


Everything in Size Town is big or small.
All big things in Size Town are heavy.
All small things in Size Town are light.
All heavy things in Size Town are still.
All light things in Size Town are unstable.
All unstable things in Size Town are changing.
All unstable things in Size Town are unpredictable.
The bird is in Size Town and it is not both heavy and still.

∀x (In(x, sizeTown) → (Big(x) ∨ Small(x)))
∀x (Big(x) ∧ In(x, sizeTown) → Heavy(x))
∀x (Small(x) ∧ In(x, sizeTown) → Light(x))
∀x (Heavy(x) ∧ In(x, sizeTown) → Still(x))
∀x (Light(x) ∧ In(x, sizeTown) → Unstable(x))
∀x (Unstable(x) ∧ In(x, sizeTown) → Changing(x))
∀x (Unstable(x) ∧ In(x, sizeTown) → Unpredictable(x))
In(bird, sizeTown) ∧ ¬(Heavy(bird) ∧ Still(bird))


The bird is either unpredictable or changing.


If the bird is small or still, then it is either unpredictable or changing.

Small(bird) ∨ Still(bird) → Unpredictable(bird) ⊕ Changing(bird)


DI Ray is a police procedural television series.
DI Ray was created and written by Maya Sondhi.
DI Ray was produced by Jed Mercurio.
Maya Sondhi and Jed Mercurio are both British.

TelevisionSeries(dIRay) ∧ PoliceProcedural(dIRay)
Creates(maya, dIRay) ∧ Writes(maya, dIRay)
Produces(jed, dIRay)
British(maya) ∧ British(jed)


∃x ∃y(British(x) ∧ TelevisionSeries(y) ∧ Produces(x, y))


Everyone who took the bar exam can read.
All lawyers took the bar exam.
Everyone who took the bar exam is knowledgeable about criminal procedures.
All people who got a score of 180 on the LSAT can read.
No elephants can read.
If Mike can not read or is not an elephant, then Mike either took the bar exam or can read.

∀x (Take(x, barExam) → CanRead(x))
∀x (Lawyer(x) → Take(x, barExam))
∀x (Take(x, barExam) → KnowledgeableAbout(x, criminalProceeder))
∀x (GetOn(x, scoreOf180, lSAT) → CanRead(x))
∀x (Elephant(x) → ¬CanRead(x))
¬(CanRead(mike) ∧ Elephant(mike)) → Take(mike, barExam) ⊕ CanRead(mike)


Mike did not take the bar exam and is not both knowledgeable about criminal procedures and someone who got 180 on the LSAT.

¬Take(mike, barExam) ∧ ¬(KnowledgeableAbout(mike, criminalProcedures)∧ GetOn(mike, 180, lSAT))


Some soccer defenders are center-backs.
All soccer defenders are soccer players.
No soccer players are professional basketball players.
All NBA players are professional basketball players.
Stephen Curry is an NBA player.

∃x (ProfessionalSoccerDefender(x) ∧ ProfessionalCenterback(x))
∀x (ProfessionalSoccerDefender(x) → ProfessionalSoccerPlayer(x))
∀x (ProfessionalSoccerPlayer(x) → ¬ProfessionalBasketballPlayer(x)))
∀x (NBAPlayer(x) → ProfessionalBasketballPlayer(x))
NBAPlayer(stephencurry)


Stephen Curry is not both a centerback and a soccer defender.


If Stephen Curry is not both a centerback and a soccer defender, then Stephen Curry is neither a soccer player nor a professional basketball player.

¬(Centerback(stephenCurry) ∧ SoccerDefender(stephenCurry)) → ¬(SoccerPlayer(stephenCurry) ∨ ProfessionalBasketballPlayer(stephenCurry))


If a person doesn't have enough money to buy a product, then that person can't buy it.
Monitors are products.
4k monitors are more expensive than 1080 monitors and 2k monitors.
John is a person.
John doesn't have enough money to buy a 2k monitor.

∀x ∀y (Person(x) ∧ Product(y) ∧ ¬HaveEnoughMoneyFor(x, y) → ¬Buy(x, y))
∀x (Monitor(x) → Product(x))
∀x ∀y ∀z (Monitor(x) ∧ Monitor(y) ∧ Monitor(z) ∧ FourK(x) ∧ OneOEightO(y) ∧ TwoK(z) → MoreExpensive(x, y) ∧ MoreExpensive(x, z))
Person(john)
∀x (Monitor(x) ∧ TwoK(x) → ¬HaveEnoughMoneyFor(john, x))

∀x (Monitor(x) ∧ OneOEightO(x) → ¬Buy(john, x))


∀x (Monitor(x) ∧ TwoK(x) → ¬Buy(john, x))


All artificial satellites are important scientific achievements.
Some artificial satellites are not U.S. inventions.

∀x (ArtificialSatellite(x) → ImportantScientificAchievement(x))
∃x (ArtificialSatellite(x) ∧ ¬USInvention(x))

All important scientific achievements are U.S. inventions.

∀x (ImportantScientificAchievement(x) ∧ USInvention(x))


Some cats are not pets.
All cats are mammals.

∃x (Cat(x) ∧ ¬Pet(x))
∀x (Cat(x) → Mammal(x))

∃x ∃y (Mammal(x) ∧ Mammal(y) ∧ ¬Pet(x) ∧ ¬Pet(y))


If people in this neighborhood visit a coffee shop regularly, then they are addicted to coffee.
People in this neighborhood visit a coffee shop regularly or order takeout at least once a day.
If people in this neighborhood make a lot of their own food at home using recipes and online guides, then they order takeout at least once a day.
If people in this neighborhood own at least one coffeemaker and one blender in their home, then they do not order takeout at least once a day.
All people in this neighborhood who lead very busy lives that include 12-hour work hours make a lot of their own food at home using recipes and online guides.
Sam is living in this neighborhood, and he is either addicted to coffee or other caffeinated drinks and leads a very busy life that include 12-hour work hours, or is not addicted to coffee and does not lead a very busy life that include 12-hour work hours

∀x (In(x, thisNeighborhood) ∧ VisitRegularly(x, coffeeShop) → AddictedTo(x, coffee))
∀x (In(x, thisNeighborhood) → (VisitRegularly(x, coffeeShop) ∨ (∃y (TakeOut(y) ∧ Order(x, y, aDay))))
∀x (In(x, thisNeighborhood) ∧ MakeAtUsing(x, home, ownFood, recipe) → ∃y (TakeOut(y) ∧ Order(x, y)))
∀x (In(x, thisNeighborhood) ∧ ∃y (CoffeemakerAndBlender(y) ∧ Own(x, y)) → ¬OrderAtLeastOnceADay(x, takeout))
∀x (In(x, thisNeighborhood) ∧ BusyWith(x, 12HourWorkHour) → MakeAtUsing(x, home, ownFood, recipe))
In(sam, thisNeighborhood) ∧ ¬(AddictedTo(sam, coffee) ⊕ BusyWith(sam, 12HourWorkHour))

Sam is living in this neighborhood and he is addicted to coffee.

InThisNeighborhood(sam) ∧ AddictedTo(sam, coffee)


Sam is living in this neighborhood and he owns at least one coffeemaker and one blender in his home.

∃y (CoffeemakerAndBlender(y) ∧ Own(sam, y))


Sam is living in this neighborhood and he owns at least one coffeemaker and one blender in his home or orders takeout at least once a day.

(∃y (CoffeemakerAndBlender(y) ∧ Own(sam, y)) ∨ (∃y (TakeOut(y) ∧ Order(sam, y, aDay)))


No professional basketball players are soccer players.
All NBA players are professional basketball players.
All soccer defenders are soccer players.
All centerbacks are soccer defenders.
Roger Federer is either both an NBA player and a soccer defender, or neither.

∀x (ProfessionalBasketballPlayer(x) → ¬ProfessionalSoccerPlayer(x))
∀x (NBAPlayer(x) → ProfessionalBasketballPlayer(x))
∀x (ProfessionalSoccerDefender(x) → ProfessionalSoccerPlayer(x))
∀x (ProfessionalCenterback(x) → ProfessionalSoccerDefender(x))
¬(NBAPlayer(rogerfederer) ⊕ ProfessionalSoccerDefender(rogerfederer))


Some teachers who work at pools are not nice.
All teachers working at pools are pool managers.
All pool managers are lifeguards.
If someone is a lifeguard, then they work at a pool.
Mary does not work at a pool.

∃x (Teacher(x) ∧ WorkAt(x, pool) ∧ ¬Nice(x))
∀x (Teacher(x) ∧ WorkAt(x, pool) → PoolManager(x))
∀x (PoolManager(x) → Lifeguard(x))
∀x (Lifeguard(x) → WorkAt(x, pool))
¬WorkAt(mary, pool)


Mary is not a nice teacher working at a pool.

¬(Nice(mary) ∧ Teacher(mary) ∧ WorkAt(mary, pool))


Mary is is a nice teacher working at a pool.

Nice(mary) ∧ Teacher(mary) ∧ WorkAt(mary, pool)


Not all art pieces require talent.
Everything that requires talent requires practice.

∃x (ArtPiece(x) ∧ ¬Require(x, talent))
∀x (Require(x, talent) → Require(x, practice))

There exist art pieces that do not require practice.

∃x ∃y (ArtPiece(x) ∧ ¬Require(x, practice) ∧ ArtPiece(y) ∧ ¬Require(y, practice) ∧ ¬(x=y))


Bernarda Bryson Shahn was a painter and lithographer.
Bernarda Bryson Shahn was born in Athens, Ohio.
Bernarda Bryson Shahn was married to Ben Shahn.
People born in Athens, Ohio, are Americans.

Painter(bernardaBrysonShahn) ∧ Lithographer(bernardaBrysonShahn)
BornIn(bernardaBrysonShahn, athensOhio)
MarriedTo(bernardaBrysonShahn, benShahn)
∀x (BornIn(x, athensOhio) → American(x))

Bernarda Bryson Shahn was born in Greece.


Bernarda Bryson Shahn had been divorced once.


Everybody in Emma's family who upgrade to the newest iPhone model every year, are not saving money for a down payment on a new house.
Everybody in Emma's family who enjoy reading about tech specs and keeping up to date on the latest technology upgrade to the newest iPhone model every year.
Everybody in Emma's family is saving money for a down payment on a new house, or lives in an apartment in a big metropolitan cities.
Everybody in Emma's family live with at least one roommate, does not own any pets.
Everybody in Emma's family who owns at least one pet lives with at least one roommate.
Emily is in Emma's family.
If Emily does not both own at least one pet and lives in apartments in big metropolitan cities, then Emily either owns at least one pet and lives in an apartment in big metropolitan cities, or she neither owns a pet nor lives in an apartment in big metropolitan cities.

∀x (InEmmasFamily(x) ∧ UpgradeToEveryYear(x, newestIphoneModel) → ¬SavingMoneyForOn(x, downPayment, newHouse))
∀x (InEmmasFamily(x) ∧ EnjoyReading(x, techSpec) ∧ KeepUpdatedOn(x, latestTechnology) → UpgradeToEveryYear(x, newestIphoneModel))
∀x (InEmmasFamily(x) ∧ (SavingMoneyForOn(x, downPayment, newHouse) ∨ LiveIn(x, apartment, bigMetropolitanCity)))
∀x (InEmmasFamily(x) ∧ (∃y (LiveWith(x, y) ∧ Roommate(y))) → ¬(∃y (Own(x, y) ∧ Pet(y))))
∀x (InEmmasFamily(x) ∧ (∃y (Own(x, y) ∧ Pet(y))) → (∃y (LiveWith(x, y) ∧ Roommate(y))))
InEmmasFamily(emily)
((∃y (Own(emily, y) ∧ Roommate(y))) ∧ LiveIn(emily, apartment, bigMetropolitanCity)) → ((∃y (Own(emily, y) ∧ Pet(y))) ∧ LiveIn(emily, apartment, bigMetropolitanCity)) ⊕ ¬((∃y (Own(emily, y) ∧ Roommate(y))) ∨ LiveIn(emily, apartment, bigMetropolitanCity))

Emily is in Emma's family and she lives with at least one roommate.


Emily enjoys reading about tech specs and keeping up to date on the latest technology.

EnjoyReading(emily, techSpec) ∧ KeepUpdatedOn(emily, latestTechnology)


People on the payroll are being paid by the school.
If someone has a job at a school, then they are on the payroll.
All faculty members have a job at a school.
If someone teaches students, they are a faculty member or a teacher.
Every teacher has students.
If Nancy is a teacher, then they are on the payroll.
If Nancy is not a teacher, then they are not paid by the school.
Nancy teaches students.

∀x (OnPayroll(x) → PaidBy(x, school))
∀x (HaveJobAt(x, school) → OnPayroll(x))
∀x (FacultyMember(x) → HaveJobAt(x, school))
∀x (Teach(x, student) → FacultyMember(x) ∨ Teacher(x))
∀x (Teacher(x) → Have(x, student))
Teacher(nancy) → OnPayroll(nancy)
¬Teacher(nancy) → ¬OnPayroll(nancy)
Teach(nancy, student)


Nancy is paid by the school and has students.

PaidBy(nancy, school) ∧ Have(nancy, student)


Nancy is not paid by the school or does not have students.

¬PaidBy(nancy, school) ∨ ¬Have(nancy, student))


Kangaroos are an animal.
No Kangaroos live in Germany.
Jane will fly to Germany if she saves enough money for the summer.
If Jane flies to Germany, she will go to the Berlin Zoo.
If someone goes to the Berlin Zoo, they will see some of the animals in Germany.

∀x (Kangaroo(x) → Animal(x))
∀x (Kangaroo(x) → ¬LiveIn(x, germany))
SavesFor(jane, enoughMoney, theSummer) → FlyTo(jane, germany)
FlyTo(jane, germany) → GoTo(jane, berlinzoo)
∀x ∃y (GoTo(x, berlinzoo) ∧ LiveIn(x, germany) ∧ Animal(y) → WillSee(y, x, berlinzoo))

Jane will see a kangaroo if she saves enough money for the summer.

∃x (SavesFor(jane, enoughMoney, theSummer)  ∧ Kangaroo(x) → WillSee(x, jane, berlinzoo))


If a class has prerequisites, the student must take the prerequisites to take the class.
If a class has no prerequisites, then the student can take the class
CPSC 201 and CPSC 223 are prerequisites for CPSC 323.
Intro Microeconomics is the only prerequisite for Intermediate Microeconomics.
Intro Geology has no prerequisites.

∀x ∀y ∀z (Class(x) ∧ Student(y) ∧ Prereq(z,x) ∧ ¬Take(y, z) → ¬CanTake(y, x))
∀x ∀y ((Class(x) ∧ Student(y) ∧ ¬∃z Prereq(z,x)) → CanTake(y, x))
Prereq(cpsc201, cpsc323) ∧ Prereq(cpsc223, cpsc323)
∀x (Prereq(x,intermediateMicro) → x=introMicroeconomics)
¬(∃x (Prereq(x, introGeology)))


If a student took CPSC 201 but did not take CPSC 223, they can take CPSC 323.

Taken(cpsc201) ∧ ¬Taken(cpsc223) ∧ CanTake(cpsc323)


Intermediate Microeconomics has one prerequisite.

∃x (Taken(x) → CanTake(intermediatemicro))


Heptalogyy is a compound literary or narrative work that is made up of seven distinct works.
The Harry Potter series consists of 7 distinct works.
The Chronicles of Narnia consists of 7 distinct works.

∀x (SevenDistinctWorks(x) → Heptalogy(x))
SevenDistinctWorks(harryPotter)
SevenDistinctWorks(chroniclesOfNarnia)

The Harry Potter series of books is Heptalogy.


The Chronicles of Narnia series of books is not Heptalogy.


All people who attend Renaissance fairs regularly enjoy dressing up in old-fashioned and historical period clothing.
If people are fascinated by the history of the Renaissance and other past eras, then they attend Renaissance fairs regularly.
People are fascinated by the history of the Renaissance and other past eras, or they are contemporary academics who enjoy learning.
People who are focused on futuristic and vocational subjects are contemporary academics who enjoy learning.
If people are professors who take a historical approach, then they are not contemporary academics who enjoy learning.
If Clyde is not focused on futuristic and voctional subjects, then he is neither focused on futuristic and vocational subjects nor enjoys dressing up in old-fashioned and historical period clothing.

∀x (AttendRegularly(x, renaissanceFair) → Enjoy(x, dressingUp, oldFashionedClothing) ∧ Enjoy(x, dressingUp, historicalPeriodClothing))
∀x (FascinatedBy(x, historyOfRenaissance) → AttendRegularly(x, renaissanceFair))
∀x (FascinatedBy(x, historyOfRenaissance) ⊕ (ContemporaryAcademic(x) ∧ Enjoy(x, learning)))
∀x (FocusedOn(x, futuristicSubject) ∧ FocusedOn(x, vocationalSubject) → ContemporaryAcademic(x) ∧ Enjoy(x, learning))
∀x (Professor(x) ∧ Take(x, historicalApproach) → ¬(ContemporaryAcademic(x) ∧ Enjoy(x, learning)))
¬(FocusedOn(clyde, futuristicSubject) ∧ FocusedOn(clyde, vocationalSubject))→ ¬(FocusedOn(clyde, futuristicSubject) ∧ FocusedOn(clyde, vocationalSubject) ∨ (Enjoy(clyde, dressingUp, oldFashionedClothing) ∧ Enjoy(clyde, dressingUp, historicalPeriodClothing)))

Clyde attends Renaissance fairs regularly.


Clyde is a professor who takes a historical approach.

Professor(clyde) ∧ Take(clyde, historicalApproach)


Clyde is a professor who takes a historical approach, or is a contemporary academic.

(Professor(clyde) ∧ Take(clyde, historicalApproach)) ∨ (ContemporaryAcademic(clyde) ∧ Enjoy(clyde, learning))


No sports cars are vehicles intended to be driven at moderate speeds.
All automobiles designed for family use are vehicles intended to be driven at moderate speeds.

∀x (SportsCar(x) → ¬IntendedToBeDrivenAt(x, moderateSpeed))
∀x (DesignedFor(x, familyUse) → IntendedToBeDrivenAt(x, moderateSpeed))

No sports cars are automobiles designed for family use.


If people work well in teams in the workplace, then they get along with all their colleagues at their work.
If people come to work every day with a positive attitude, then they work well in teams in the workplace.
People either come to work every day with a positive attitude or are always tired every morning.
If people are always tired in the morning, then they are criticized by their boss.
If people are criticized by their boss, then they do not receive positive feedback from teams at work.
Kat either is a person who works well in teams in the workplac and is always tired every morning, or she is neither.

∀x (WorkWellInTeamsIn(x, workPlace) → ∀y (Colleague(y) ∧ GetAlongWithAtWork(x, y)))
∀x (ComeToWorkWithEveryDay(x, positiveAttitude) → WorkWellInTeamsIn(x, workPlace))
∀x (ComeToWorkWithEveryDay(x, positiveAttitude) ⊕ AlwaysTiredInMorning(x))
∀x (AlwaysTiredInMorning(x) → CriticizedBy(x, boss))
∀x (CriticizedBy(x, boss) → ¬ReceiveFromAtWork(x, positiveFeedback, team))
¬(WorkWellInTeamsIn(kat, workPlace) ⊕ Tired(kat))

Kat is a person who comes to work every day with a positive attitude.


Kat gets along with her colleagues at her work and receives positive feedback from teams at her work.

(∀y (Colleague(y) ∧ GetAlongWithAtWork(kat, y))) ∧ ReceiveFromAtWork(kat, positiveFeedback, team)


Kat either gets along with her colleagues at her work or receives positive feedback from teams at her work.

(∀y (Colleague(y) ∧ GetAlongWithAtWork(kat, y))) ⊕ ReceiveFromAtWork(kat, positiveFeedback, team)


Drishti is an open-source software.
Open-source software is free to modify.

OpenSourceSoftware(drishti)
∀x (OpenSourceSoftware(x) → FreeToModify(x))


There are five grades in English class: A+, A, B+, B, and C.
If a student gets an A+ in English class, then his score is greater than 95.
If a student gets an A in English class, then his score is greater than 90 but lower than 95.
Zhang got an A in English class.
Wang's English class score is better than Zhang's.
Wu's English class score is lower than 90.
If a student's English class score is lower than 90, then it is not greater than 95 or 90, and lower than 95.

GradeIn(aPlus, englishClass) ∨ GradeIn(a, englishClass) ∨ GradeIn(bPlus, englishClass) ∨ GradeIn(b, englishClass) ∨ GradeIn(c, englishClass) ∧ (GradeIn(aPlus, englishClass) → ¬GradeIn(a, englishClass) ∧ ¬GradeIn(bPlus, englishClass) ∧ ¬GradeIn(b, englishClass) ∧ ¬GradeIn(c, englishClass)) ∧ (GradeIn(a, englishClass) → ¬GradeIn(aPlus, englishClass) ∧ ¬GradeIn(bPlus, englishClass) ∧ ¬GradeIn(b, englishClass) ∧ ¬GradeIn(c, englishClass)) ∧ (GradeIn(bPlus, englishClass) → ¬GradeIn(aPlus, englishClass) ∧ ¬GradeIn(a, englishClass) ∧ ¬GradeIn(b, englishClass) ∧ ¬GradeIn(c, englishClass)) ∧ (GradeIn(b, englishClass) → ¬GradeIn(aPlus, englishClass) ∧ ¬GradeIn(a, englishClass) ∧ ¬GradeIn(bPlus, englishClass) ∧ ¬GradeIn(c, englishClass)) ∧ (GradeIn(c, englishClass) → ¬GradeIn(aPlus, englishClass) ∧ ¬GradeIn(a, englishClass) ∧ ¬GradeIn(bPlus, englishClass) ∧ ¬GradeIn(b, englishClass))
∀x ∀y (Student(x) ∧ GetGradeIn(x, aPlus, englishClass) → EnglishClassScore(x, y) ∧ GreaterThan95(y))
∀x ∀y (Student(x) ∧ GetGradeIn(x, a, englishClass) → EnglishClassScore(x, y) ∧ GreaterThan90(y) ∧ LowerThan95(y))
Student(zhang) ∧ GetGradeIn(zhang, a, englishClass)
∀x ∀y (Student(zhang) ∧ Student(wang) ∧ EnglishScore(zhang, x) ∧ EnglishScore(wang, y) ∧ Better(y, x))
∀x (Student(wu) ∧ EnglishScore(wu, x) ∧ LowerThan90(x))
∀x ∀y (Student(x) ∧ EnglishScore(x, y) ∧ LowerThan90(y) → ¬GreaterThan95(y) ∧ ¬GreaterThan90(y) ∧ LowerThan95(y))

Zhang's English class score is lower than 95.

∀x (EnglishScore(zhang, x) ∧ LowerThan95(x))


Wu does not get an A or A+ in English class.

¬GetGradeIn(wu, aPlus, englishClass) ∧¬GetGradeIn(wu, a, englishClass)


Olivia doesn't prefer warm temperatures during the day.
When Olivia sleeps, she prefers a cool temperature.
Olivia sleeps during the night.
Olivia works during the day.
Olivia either works or sleeps.
It is either the day or the night.
Olivia either prefers warm temperatures or prefers cool temperatures.

∀x (Day(x) → ¬Prefer(olivia, warmTemperature, x))
∀x (Sleep(olivia, x) → Prefer(olivia, coolTemperature, x))
∀x (Night(x) → Sleep(olivia, x))
∀x (Day(x) → Work(olivia, x))
Work(olivia) ⊕ Sleep(olivia)
∀x (Day(x) ⊕ Night(x))
∀x (Prefer(olivia, warmTemperature, x) ⊕ Prefer(olivia, coolTemperature, x))

At all times, Olivia prefers a cool temperature.


TOra is a GUI.
GUIs are software.
Software can be free or paid.
Paid Software is not under the GNU General Public License.
TOra is under the GNU General Public License.

GUI(tora)
∀x (GUI(x) → Software(x))
∀x (Software(x) → Free(x) ⊕ Paid(x))
∀x (Paid(x) ∧ Software(x) → ¬UnderGNULicense(x))
UnderGNULicense(tora)


Customers choose a Prime Video plan or an HBO Max Plan, or both.
All customers who choose a Prime Video Plan are rewarded with a $30 gift card.
There are no customers who do not choose any plan.
None of the customers who are rewarded with a $30 gift card are older than 80.
All the customers are either older than 80 or between the ages of 60 and 80.
James is a customer who is not between the ages of 60 and 80.

∀x (Customer(x) → (Choose(x, primeVideoPlan) ∨ Choose(x, hBOMaxPlan)))
∀x ((Customer(x) ∧ Choose(x, hBOMaxPlan)) → RewardWith(x, giftCard))
∀x (Customer(x) → (∃y(Plan(y) ∧ Choose(x, y))))
∀x ((Customer(x) ∧ RewardWith(x, giftCard)) → (¬OlderThan(x, num80)))
∀x (Customer(x) → (∃y(GreaterThan(y, num80) ∧ Age(james,y)) ⊕ (∃y(Between(y, num60, num80) ∧ Age(james, y)))))
Customer(james) ∧ (¬∃y(Between(y, num60, num80) ∧ Age(james, y)))

James is a customer who does not choose any plans.


James is a customer who chooses a Prime Video plan or does not choose any plans.

Choose(james, planA) ∨ Choose(james, noPlan)


Suppose James is a customer who chooses the Prime Video plan or does not choose any plans, then he is either rewarded a $30 gift card or chooses the HBO Max plan.

Choose(james, planA) ∨ Choose(james, noPlan) → RewardWith(james, giftCard) ⊕ Choose(james, planB)


Detroit City is a horse.
Some horses are racehorses.
If a horse falls in a race, it poses risks to its rider.
Detroit City fell in a race.
A horse is a racehorse if it is in a race.

Horse(detroitcity)
∃x (Horse(x) ∧ Racehorse(x))
∀x (Horse(x) ∧ InRace(x) ∧ Falls(x) → PoseRiskTo(x, rider))
InRace(detroitcity) ∧ Fall(detroitcity)
∀x (Horse(x) ∧ InRace(x) → Racehorse(x))

Detroit City has been in multiple races.


Frederick Monhoff was an architect, artist, and illustrator.
Frederick Monhoff was an American.
An artist is good at physical or conceptual art.
All Americans are American citizens.

Architect(monhoff) ∧ Artist(monhoff) ∧ Illustrator(monhoff)
American(monhoff)
∀x (Artist(x) → GoodAt(x, physicalArt) ∨ GoodAt(x, conceptualArt))
∀x (American(x) → AmericanCitizen(x))

Frederick Monhoff was good at physical art.


¬(∃x (Illustrator(x) ∧ AmericanCitizen(x)))


Miroslav Fiedler was a Czech mathematician.
Miroslav Fiedler is known for his contributions to linear algebra and graph theory.
Miroslav Fiedler is honored by the Fiedler eigenvalue.
Fiedler eigenvalue is the second smallest eigenvalue of the graph Laplacian.

Czech(miroslavFiedler) ∧ Mathematician(miroslavFiedler)
KnownFor(miroslavFiedler, contributionsToLinearAlgebraAndGraphTheory)
HonoredBy(miroslavFiedler, fiedlerEigenvalue)
TheSecondSmallestEigenvalueOf(fiedlerEigenvalue, theGraphLaplacian)

Miroslav Fiedler is honored by the second smallest eigenvalue of the graph Laplacian.

∃x (TheSecondSmallestEigenvalueOf(x, theGraphLaplacian) ∧ HonoredBy(miroslavFiedler, x))


Miroslav Fiedler was a French mathematician.


A Czech mathematician is known for his contributions to linear algebra and graph theory.

∃x (Czech(x) ∧ Mathematician(x) ∧ KnownFor(x, contributionsToLinearAlgebraAndGraphTheory))


A laptop is a computer.
You can play games on a computer.
A phone is not a computer.

∀x (Laptop(x) → Computer(x))
∀x (Computer(x) → CanPlayGameOn(x))
∀x (Phone(x) → ¬Computer(x))


Walter Folger Brown was an American politician and lawyer who served as the postmaster general.
Walter Folger Brown graduated from Harvard University with a Bachelor of Arts.
While they were both in Toledo, Walter Folger Brown's father practiced law with Walter Folger Brown.
Katherin Hafer married Walter Folger Brown.

AmericanPolitician(walterBrown) ∧ Lawyer(walterBrown) ∧ ServedAs(walterBrown, postMasterGeneral)
Graduated(walterBrown, harvard) ∧ GraduatedWith(walterBrown, bachelorsOfArt)
∃t(In(walterBrown, toledo, t) ∧ In(walterBrownFather, toledo, t) ∧ PracticedLawTogether(walterBrown, walterBrownFather, t))
Married(katherinHafer, walterBrown)

Walter Folger Brown graduated with a Bachelor of Arts.


Walter Folger Brown's father was in Toledo.


All products designed by Apple are sold at Apple Stores.
All products with Apple logos are designed by Apple.
All Macbooks have Apple logos.
All products with Apple M2 chips are Mackbooks.
A Thinkpad X1 is not both sold in Apple Stores and is a Macbook.

∀x ((Product(x) ∧ DesignedBy(x, apple)) → SoldIn(x, appleStore))
∀x ((Product(x) ∧ With(x, appleLogo)) → DesignedBy(x, apple))
∀x (Macbook(x) → With(x, appleLogo))
∀x ((Product(x) ∧ With(x, appleM2Chip)) → Macbook(x))
¬(SoldIn(thinkpadX1, appleStore) ∧ Macbook(thinkpadX1))


The Thinkpad X1 is sold in Apple Stores.


The Thinkpad X1 has an Apple M2 chip and is a Macbook.

With(thinkpadX1, appleM2Chip) ∧ Macbook(thinkpadX1)


The Thinkpad X1 either has an Apple M2 chip or is a Macbook.

With(thinkpadX1, appleM2Chip)) ⊕ Macbook(thinkpadX1)


If the Thinkpad X1 has an Apple M2 chip and is a Macbook, then it neither has an Apple M2 chip nor is sold in Apple Stores.

(With(thinkpadX1, appleM2Chip) ∧ Macbook(thinkpadX1)) → ¬(With(thinkpadX1, appleM2Chip) ∨ SoldIn(thinkpadX1, appleStore))


Oxford Circus is a road junction connecting Oxford Street and Regent Street.
Oxford Street and Regent Street are in London.
John Nash designed a construction on Regent Street.
John Nash designed Oxford Circus.
John Nash is a British architect.
Oxford Circus is the entrance to Oxford Circus tube station, a part of the Central line in 1900.

RoadJunction(oxfordCircus) ∧ Connect(oxfordCircus, oxfordSt, regentSt)
In(oxfordSt, london) ∧ In(regentSt, london)
Designed(nash, construction) ∧ On(construction, regentSt)
Designed(nash, oxfordCircus)
Architect(nash) ∧ British(nash)
EntraceTo(oxfordCircus, tubeStation) ∧ PartOf(tubeStation, centralline) ∧ In(tubeStation, 1900)


Oxford Circus is designed by a British architect.

∃x (British(x) ∧ Architect(x) ∧ Design(x, oxfordCircus))


John Nash designed the Central line in 1900.

∀x (PartOf(x, centralLine) → Design(johnNash, x))


All pets in my house are either cats or dogs.
All the dogs in my house bark.
Ghosts do not exist.
If some pet in my house barks, then it is not dead.
All of the pets in my house are either dead or alive.
Jojo is a pet in my house, and it is not alive.

∀x (Pet(x) ∧ In(x, myHouse) → Cat(x) ⊕ Dog(x))
∀x (Dog(x) ∧ In(x, myHouse) → Bark(x))
∀x (¬Ghost(x))
∀x (Bark(x) ∧ Pet(x) ∧ In(x, myHouse) → ¬Dead(x))
∀x (Pet(x) ∧ In(x, myHouse) → Dead(x) ⊕ Alive(x))
Pet(jojo) ∧ InMyHouse(jojo)∧ ¬Alive(jojo)


If Jojo is a cat or a ghost, then Jojo either barks or is a dog.

Cat(jojo) ∨ Ghost(jojo) → Bark(jojo) ⊕ Dog(jojo)


All tigers are cats.
No cats are dogs.
All Bengal tigers are tigers.
All huskies are dogs.
Fido is either a Bengal tiger or a cat.

∀x (Tiger(x) → Cat(x))
∀x (Cat(x) → ¬Dog(x))
∀x (BengalTiger(x) → Tiger(x))
∀x (Husky(x) → Dog(x))
BengalTiger(fido) ⊕ Cat(fido)


If a city holds a Summer Olympics, and the city is a US city, then the Summer Olympics will be in the US.
If a city is in a state in the US, the city is a US city.
If a city is in a state, and a Summer Olympics is in this city, then the Summer Olympics is in this state.
The 2028 Summer Olympics is scheduled to take place in Los Angeles.
Los Angeles is a city in California.
Atlanta is a US city.
Atlanta is in Georgia.
California is a state in the United States.
Boxing, modern pentathlon, and weightlifting will be removed from The 2028 Summer Olympics.
Atlanta in the United States held the 1996 Summer Olympics.

∀x ∀y ((SummerOlympicsIn(x,y) ∧ In(x, unitedStates)) → SummerOlympicsIn(x, unitedStates))
∀x ∀y ((In(x, y) ∧ In(y, unitedStates)) → In(x, unitedStates))
∀x ∀y ∀z ((In(x, z) ∧ State(z) ∧ SummerOlympicsIn(x,y)) → SummerOlympicsIn(z, y))
SummerOlympicsIn(losAngeles, yr2028)
In(losAngeles, california)
In(atlanta, unitedStates)
In(california, unitedStates)
In(atlanta, georgia)
¬InSummerOlympicsIn(boxing, yr2028) ∧ (¬InSummerOlympicsIn(modern_pentathlon, yr2028)) ∧ (¬InSummerOlympicsIn(weightlifting, yr2028))
SummerOlympicsIn(atlanta, yr1996)

The 2028 Summer Olympics will take place in the US.


The 1996 Summer Olympics is not in Georgia.


Skateboarding will appear at The 2028 Summer Olympics.


The taiga vole is a large vole found in northwestern North America.
Cats like playing with all voles.
The taiga vole lives in the boreal taiga zone.
The boreal taiga zone in North America is a cold place to live in.

Vole(taigaVole) ∧ LiveIn(taigaVole, northAmerica)
LikePlayingWith(cat, taigaVole)
LiveIn(taigaVole, borealTaigaZone)
∀x ((LiveIn(x, northAmerica) ∧ LiveIn(x, borealTaigaZone)) → LiveIn(x, coldPlace))


A diseases affect females or males.
No women have prostate cancer.
A cancer is either prostate cancer or non-prostate cancer.
No type of cancer is without mutations.
All non-prostate cancers are a type of cancer.
If adenocarcinoma is a type of cancer or without mutations or both, then adenocarcinoma is in women or without mutations or both.

∀x (Disease(x) → (Affects(x, female) ∨ Affects(x, male)) )
∀x (Affect(x, female) → ¬ProstateCancer(x))
∀x (ProstateCancer(x) ∨ NonProstateCancer(x))
∀x (Cancer(x) → ¬Without(x, mutation))
∀x (NonProstateCancer(x) → Cancer(x))
(Cancer(adenocarcinoma) ∨ Without(adenocarcinoma, mutation)) → (Affect(adenocarcinoma, female) ∨ Without(adenocarcinoma, mutation))


If adenocarcinoma is a disease in women or without mutations, then adenocarcinoma is without mutations and a non-prostate cancer.

(Affect(adenocarcinoma, men) ∨ Without(adenocarcinoma, mutation)) → (NonProstateCancer(adenocarcinoma) ∧ Without(adenocarcinoma, mutation))


Some monitors equipped in the lab are produced by the company named AOC.
All monitors equipped in the lab are cheaper than their original prices.
If a monitor is cheaper than its original price, then its resolution is 1080p.
If a monitor has a resolution of 1080p, then it does not support the type-c port.
LG34 is equipped in the lab.

∃x ∃y  (LabMonitor(x) ∧ AOC(x) ∧ (¬(x=y)) ∧ LabMonitor(y) ∧ AOC(y))
∀x (LabMonitor(x) → Discounted(x))
∀x (Discounted(x) → A1080p(x))
∀x (A1080p(x) → ¬TypeC(x))
LabMonitor(lg-34)


LG34 machine does not support the type-c port.


All fruits sold at Nica's market are shipped from Colombia.
Some fruits sold in New Haven are shipped from Mexico.
No fruits shipped from Colombia are sold at the local farmers market in New Haven.
Avocados are a kind of fruit sold at the local farmers market in New Haven or at Nica's market.
Avocados are either shipped from Colombia and sold in New Haven, or neither.

∀x ((Fruit(x) ∧ SoldAt(x, nicasMarket)) → ShippedFrom(x, colombia))
∃x ∃y (Fruit(x) ∧ SoldIn(x, newHaven) ∧ ShippedFrom(x, mexico) ∧ (¬(x=y)) ∧ Fruit(y) ∧ SoldIn(y, newHaven) ∧ ShippedFrom(y, mexico))
∀x ((Fruit(x) ∧ ShippedFrom(x, colombia)) → ¬(SoldAt(x, localFarmersMarket)))
Fruit(avocado) ∧ (SoldAt(avocado, localFarmersMarket) ∨ SoldAt(avocado, nica'sMarket))
¬(ShippedFrom(avocado, colombia) ⊕ SoldIn(avocado, newHaven))

Avocados are a kind of fruit sold at the local farmers market in New Haven.

Fruit(avocado) ∧ SoldAt(avocado, localFarmersMarket)


Avocados are either sold at the local farmers market in New Haven or are sold in New Haven.

SoldAt(avocado, localFarmersMarket) ⊕ SoldIn(avocado, newHaven)


Avocados are either sold in New Haven or sold at Nica's market.

SoldIn(avocado, newHaven) ⊕ SoldAt(x, nica'sMarket)


If avocados are not both sold at the local farmers market in New Haven and shipped from Columbia, then they are neither sold at the local farmers market in New Haven nor in New Haven generally.

¬(SoldAt(avocado, localFarmersMarket) ∧ ShippedFrom(avocado, colombia)) → ¬SoldAt(avocado, localFarmersMarket) ∧ ¬SoldIn(avocado, newHaven)


Some monitors equipped in the library are produced by AOC.
All monitors equipped in the library are cheaper than 800 dollars.
All monitors cheaper than 800 dollars are with a resolution lower than 1080p.
If a monitor has a resolution lower than 1080p, then it does not support the type-c port.
A-2017 supports the type-c port.

∃x ∃y(Monitor(x) ∧ ProducedBy(x, aOC) ∧ In(x, library) ∧ (¬(x=y)) ∧ Monitor(y) ∧ ProducedBy(y, aOC) ∧ In(y, library))
∀x ((Monitor(x) ∧ In(x, library)) → CheaperThan(x, dollars800))
∀x ((Monitor(x) ∧ CheaperThan(x, dollars800)) → ResolutionLessThan(x, p1080))
∀x ((Monitor(x) ∧ ResolutionLessThan(x, p1080)) → ¬Supports(x, type-CPort))
Supports(a-2017, type-CPort)


A-2017 is produced by AOC and equipped in the library.

ProducedBy(a-2017, aOC) ∧ In(a-2017, library)


If either A-2017 is both with a resolution of 1080p and produced by AOC or it is neither, then it is not equipped in the library.

¬(ResolutionLessThan(a-2017, p1080) ⊕ ProducedBy(x, aOC)) → ¬(In(a-2017, library))


Sūduva Marijampolė holds the Lithuanian Super Cup.
Sūduva Marijampolė is a soccer team.

Some soccer team holds the Lithuanian Super Cup.

∃x (SoccerTeam(x) ∧ Holds(x, theLithuanianSuperCup))


Ainderby Quernhow is a village and civil parish in the Hambleton District.
Hambleton District is in North Yorkshire.
North Yorkshire is in England.
If place A is located in place B and place B is located in place C, then place A is located in place C.

Village(ainderbyQuernhow) ∧ CivilParish(ainderbyQuernhow) ∧ In(ainderbyQuernhow, hambletonDistrict)
In(hambletonDistrict, northYorkshire)
In(northYorkshire, england)
∀x ∀y ∀z ((In(x, y) ∧ In(y, z)) → In(x, z))


Douglas Adams is an author who created the book collection called The Salmon of Doubt.
The Salmon of Doubt is about life experiences and technology.
All authors are writers.
Writers create innovative ideas.
Some books that contain innovative ideas are about technology.

Author(douglasAdams) ∧ Authored(douglasAdams, theSalmonOfDoubt) ∧ Book(theSalmonOfDoubt)
About(theSalmonOfDoubt, lifeExperience) ∧ About(theSalmonOfDoubt, technology)
∀x (Author(x) → Writer(x))
∀x (Writer(x) → Create(x, innovativeIdea))
∃x ∃y (Contain(x, innovativeIdea) ∧ About(x, technology) ∧ (¬(x=y)) ∧ (Contain(y, innovativeIdea) ∧ About(y, technology)))


The Salmon of Doubt has no innovative Ideas.


No disposable products can help slow down global warming.
All eco-friendly brands can help slow down global warming.
All sustainable fashion brands are eco-friendly brands.
All fast fashion products are disposable products.
If Reformation is not helping slow down global warming, then Reformation is an eco-friendly brand or a sustainable fashion brand.

∀x (Disposable(x) ∧ Product(x) → ¬HelpSlowDown(x, globalWarming))
∀x (EcoFriendly(x) ∧ Brand(x) → Help(x, slowDownGlobalWarming))
∀x (Sustainable(x) ∧ FashionBrand(x) → EcoFriendly(x) ∧ Brand(x))
∀x (FastFashion(x) ∧ Product(x) → Disposable(x) ∧ Product(x))
¬HelpSlowDown(reformation, globalWarming) → (EcoFriendly(reformation) ∧ Brand(reformation)) ∨ (Sustainable(reformation) ∧ FashionBrand(reformation))


Reformation produces fast fashion products.


Reformation does not produce fast fashion products.


Reformation does not produce fast fashion products or does not produce disposable products.

¬(FastFashion(reformation) ∧ Product(reformation)) ∨ ¬(Disposable(x) ∧ Product(x))


If Reformation produces disposable products, then Reformation produces fast fashion products.

(Disposable(reformation) ∧ Product(reformation)) → (FastFashion(reformation) ∧ Product(reformation))


If Reformation produces fast fashion products or helps slow down global warming, then Reformation produces fast fashion products.

(FastFashion(reformation) ∧ Product(reformation)) ∨ ¬HelpSlowDown(reformation, globalWarming)


Roy Richardson was a cricketer who played for Sint Maarten, a constituent country.
Roy Richardson was a right-handed batsman and medium-pace bowler.
Roy Richardson was old when he debuted in cricket.
Sherville Huggins dismissed Roy Richardson.

Cricketeer(royRichardson) ∧ PlaysFor(royRichardson, sintMaarten) ∧ ConstituentCountry(sintMaarten)
RightHanded(royRichardson) ∧ Batsman(royRichardson) ∧ MediumPaceBowler(royRichardson)
OldAtDebut(royRichardson)
Dismisses(shervilleHuggins, royRichardson)

Sherville Huggins has never dismissed anyone playing cricket for a constituent country.

∀x ∀y ((ConsituentCountry(y) ∧ PlayedFor(x, y)) →  ¬Dismissed(shervillehuggins, x))


No right-handed medium-pace bowlers were playing for Sint Maarten.

∀x ((RightHanded(x) ∧ MediumPaceBowler(x)) → ¬PlayedFor(x, sintMaarten))


To get a job at Google, you need to have a lot of work experience or a good education.
One needs to submit their resume to Google to get a job there.
John has a lot of work experience.
John submitted his resume to Google and got a job there.

∀x (GetAJobAt(x, google) → Have(x, aLotOfWorkExperience) ∨ Have(x, goodEducation))
∀x (GetAJobAt(x, google) → Submitted(x, resume, google))
Have(john, aLotOfWorkExperience)
Submitted(john, resume, google) ∧ GetAJobAt(john, google)


No iPhones are standalone desktops.
All Apple-made cellphones are iPhones.
All phones with A15 Bionic chips are Apple-made cell phones.
All phones equipped with four core-GPU made by Apple are phones with A15 Bionic chips.
If an unannounced Huawei phone is either a phone with A15 Bionic chips or equipped with four core-GPU made by Apple, then unannounced Huawei phone is neither a phone with A15 Bionic chips nor a standalone desktop.

∀x (IPhone(x) → ¬StandaloneDesktop(x))
∀x (AppleMade(x) ∧ Cellphone(x) → IPhone(x))
∀x (Phone(x) ∧ With(x, a15BionicChip) → AppleMade(x) ∧ Cellphone(x))
∀x (Phone(x) ∧ EquippedWith(x, fourCoreGPU) ∧ MadeBy(x, apple) → Phone(x) ∧ With(x, a15BionicChip))
(Phone(unannouncedHuaweiPhone) ∧ With(unannouncedHuaweiPhone, a15BionicChip)) ⊕ (Phone(unannouncedHuaweiPhone) ∧ EquippedWith(unannouncedHuaweiPhone, fourCoreGPU) ∧ MadeBy(unannouncedHuaweiPhone, apple)) → ¬(Phone(unannouncedHuaweiPhone) ∧ With(unannouncedHuaweiPhone, a15BionicChip) ∧ StandaloneDesktop(unannouncedHuaweiPhone))


Phone(unannouncedHuaweiPhone) ∧ EquippedWith(unannouncedHuaweiPhone, fourCoreGPU) ∧ MadeByApple(unannouncedHuaweiPhone)


¬(Phone(unannouncedHuaweiPhone) ∧ EquippedWith(unannouncedHuaweiPhone, fourCoreGPU) ∧ MadeByApple(unannouncedHuaweiPhone))


Hugh Vanstone is one of the world's leading lighting designers.
Hugh Vanstone is from the UK.
Hugh Vanstone has lit more than 160 productions.
Hugh Vanstone attended a school where he is from.

WorldLeadingLightingDesigner(hughVanstone)
From(hughVanstone, unitedKingdom)
∃x(GreaterThan(x, num160) ∧ LitProductions(hughVanstone,x))
∃x(Hometown(hughVanstone,x) ∧ AttendedSchoolIn(hughVanstone,x))

Hugh Vanstone is one of the world's leading lighting designers and is from the UK.

WorldLeadingLightingDesigner(hughVanstone) ∧ From(hughVanstone, unitedKingdom)


∃x(GreaterThan(x, num170) ∧ LitProductions(hughVanstone,x))


Hugh Vanstone attended a school in the United States.


No man can run faster than Bolt.
Superman is not a man.

∀x (Man(x) → ¬RunFasterThan(xm bolt))
¬Man(superman)


Donald Ervin Knuth is an American computer scientist, mathematician, and Professor Emeritus at Stanford University.
Knuth has been called the "father of the analysis of algorithms."

American(donaldErvinKnuth) ∧ ComputerScientist(donaldErvinKnuth) ∧ Mathematician(donaldErvinKnuth) ∧ ProfessorEmeritusAt(donaldErvinKnuth, stanford)
Called(donaldErvinKnuth, fatherOfTheAnalysisOfAlgorithms)

An American scientist has been called the "father of the analysis of algorithms".

∃x (American(x) ∧ ComputerScientist(x) ∧ Called(x, fatherOfTheAnalysisOfAlgorithms))


A mathematician has been called the "father of the analysis of algorithms".

∃x (Mathematician(x) ∧ Called(x, fatherOfTheAnalysisOfAlgorithms))


Donald Knuth is a well-known figure in the field of artificial intelligence.


Neocrepidodera Corpulentas are flea beetles or moths, or both.
Neocrepidodera Corpulentas are in the Chrysomelidae family.
There are no moths within the Chrysomelidae family.
There is a Neocrepidodera Corpulenta.

∀x (NeocrepidoderaCorpulenta(x) → (FleaBeetle(x) ∨ Moth(x)))
∀x (NeocrepidoderaCorpulenta(x) → In(x, chrysomelidaeFamily))
∀x (In(x, chrysomelidaeFamily) → ¬Moth(x))
∃x (NeocrepidoderaCorpulenta(x))

There is a flea beetle within the Chrysomelidae family.

∃x (FleaBeetle(x) ∧ In(x, chrysomelidaeFamily))


There are no flea beetles within the Chrysomelidae family.

∀x (FleaBeetle(x) → ¬In(x, chrysomelidaeFamily))


Carrozzeria Colli is a Milanese coachbuilder company established by Giuseppe Colli in 1931.
Carrozzeria Colli is a company that specializes in using aluminum.
The first automobiles built by Carrozzeria Colli were racing cars.
Some racing cars built by Carrozzeria Colli used Fiat 1100 mechanicals and chassis.
Carrozzeria Colli worked for airforces.
Carrozzeria Colli made car bodies.

Milanese(carrozzeriaColli) ∧ CoachBuilder(carrozzeriaColli) ∧ Company(carrozzeriaColli) ∧ EstablishedBy(carrozzeriaColli, giuseppeColli) ∧ EstablishedIn(carrozzeriaColli, 1931)
Company(carrozzeriaColli) ∧ SpecializesIn(carrozzeriaColli, usingAluminum)
∀x (BuiltBy(x, carrozzeriaColli) ∧ FirstAutomobile(x) → RacingCar(x))
∃x (BuiltBy(x, carrozzeriaColli) ∧ RacingCar(x) ∧ Used(x, fiat1100mechanicals) ∧ Used(x, chassis))
∃x (Airforce(x) ∧ WorkedFor(carrozzeriaColli, x))
∃(CarBody(x) ∧ Made(x, carrozzeriaColli))

Carrozzeria Colli made car bodies in 1931.

∃x (CarBody(x) ∧ Made(x, carrozzeriaColli) ∧ MadeIn(x, 1931))


Carrozzeria Colli built airplanes during World War II.

∃x (Airplane(x) ∧ Made(x, carrozzeriaColli) ∧ MadeDuring(x, worldWarII))


Giuseppe Colli established a company that made car bodies.

∃x ∃y (Company(x) ∧ EstablishedBy(x, giuseppeColli) ∧ CarBody(y) ∧ Made(y, x))


Giuseppe Colli established a Milanese coachbuilder company that specialized in using aluminum.

∃x (Milanese(x) ∧ CoachBuilder(x) ∧ Company(x) ∧ EstablishedBy(x, giuseppeColli) ∧ SpecializesIn(x, usingAluminum))


The first automobiles built by Carrozzeria Colli were built using Fiat 1100 mechanicals and chassis.

∃x (BuiltBy(x, carrozzeriaColli) ∧ FirstAutomobil(x) ∧ Used(x, fiat1100mechanicals) ∧ Used(x, chassis))


John will go to the cinema if and only if Jack goes to the cinema today.
Jack will go to the cinema if and only if Iron Man is on and the weather is not bad today.
Some days in March have bad weather.
Iron Man is on.
It's March now.

(GoTo(john, theCinema) ∧ GoTo(john, today)) ↔ GoTo(jack, theCinema) ∧ GoTo(jack, today)
(GoTo(john, theCinema) ∧ GoTo(john, today)) ↔ (On(ironman) ∧ ¬Bad(weather, today))
∃x (Day(x) ∧ March(x) → ¬Bad(weather, x))
On(ironman)
Day(presentMoment) ∧ March(presentMoment)


Quiksilver sells sportswear, clothing, footwear, and accessories.
Flannels are a type of clothing.
Joe owns an item from Quiksilver.

∀x (Sells(quiksilver, x) → (Sportswear(x) ∨ Clothing(x) ∨ Footwear(x) ∨ Accessory(x)))
Clothing(flannel)
∃x (Sells(quiksilver, x) ∧ Owns(joe, x))


Joe owns at least one piece of sportswear, clothing, footwear, or accessory

∃x (Owns(joe, x) ∧ Sportswear(x) ∨ Clothing(x) ∨ Footwear(x) ∨ Accessory(x))


No video games released by Nintendo support the PS4 platform.
All video games in the Pokemon series are released by Nintendo.
All video games in the FIFA series support the PS4 platform.
All video games that allow users to simulate playing online soccer using licensed players are in the FIFA series.
The video game named “Be Lionel” is in the Pokemon series, or it allows users to simulate playing online soccer games using licensed players.

∀x (VideoGame(x) ∧ ReleasedBy(x, nintendo) → ¬Support(x, pS4))
∀x (VideoGame(x) ∧ In(x, pokemonSeries) → ReleasedBy(x, nintendo))
∀x (VideoGame(x) ∧ In(x, fIFASeries) → Support(x, pS4))
∀x (VideoGame(x) ∧ Simulate(x, onlineSoccer) ∧ Use(x, licensedPlayer) → In(x, fIFASeries))
VideoGame(beLionel) ∧ In(beLionel, pokemonSeries) ∨ (Simulate(beLionel, onlineSoccer) ∧ Use(beLionel, licensedPlayer))

The video game "Be Lionel" is in the pokemon series.


The video game named “Be Lionel” either is in the FIFA series and supports the PS4 platform, or it neither is in the FIFA series nor supports the PS4 platform.

VideoGame(beLionel) ∧  ¬(FIFASeries(beLionel) ⊕ Support(beLionel, pS4))


The video game named “Be Lionel” is either in the FIFA series or supports the PS4 platform.

VideoGame(beLionel) ∧ FIFASeries(beLionel) ⊕ Support(beLionel, pS4)


The video game named “Be Lionel”  is not in the FIFA or Pokemon series.

VideoGame(beLionel) ∧ ¬(FIFASeries(beLionel) ∨ Support(beLionel, pS4))


No payment cards issued by Russian banks can be used with ApplePay.
All MIR payment cards are issued by Russian banks.
Some international payment cards can be used with ApplePay.
Social payments in Russia can only be transferred to MIR payment cards.
Bank of America payment cards can be used with ApplePay.

∀x ∀y (PaymentCard(x) ∧ RussianBank(y) ∧ IssuedBy(x, y) → ¬UsedWith(x, applePay))
∀x ∀y (PaymentCard(x) ∧ MIR(x) → RussianBank(y) ∧ IssuedBy(x, y))
∃x (PaymentCard(x) ∧ International(x) → UsedWith(x, applePay))
∀x ∀y (SocialPayment(x) ∧TransferredTo(x, y) → PaymentCard(y) ∧ MIR(y))
PaymentCard(bankOfAmerica) ∧ UsedWith(bankOfAmerica, applePay)

Bank of America payment cards are international.


Bank of America payment cards are international and can be used to transfer social payments in Russia.

∀x (PaymentCard(bankOfAmerica) ∧ International(bankOfAmerica) ∧ SocialPayment(x) ∧TransferredTo(x, bankOfAmerica))


If Bank of America payment cards are international and can be used to transfer social payments in Russia, then they are international.

∀x ((PaymentCard(bandOfAmerica) ∧ International(bandOfAmerica) ∧ SocialPayment(x) ∧ TransferredTo(x, bandOfAmerica)) → International(bandOfAmerica))


The Lumina APV is produced by Chevrolet.
The Astro is a van produced by Chevrolet.
Vehicles produced by Chevrolet in this batch are either cars or vans.

ProducedBy(luminaAPV, chevrolet)
ProducedBy(astro, chevrolet) ∧ Van(astro)
∀x (Vehicle(x) ∧ ProducedBy(x, chevrolet) ∧ InThisBatch(x) → (Car(x) ⊕ Van(x)))


The Lumina APV is either a car or a van.


Everyone who works in the office is a commuter.
People either work in the office or work from home.
Everyone who works from home has a relaxed schedule.
George is either a commuter or has a home office setup.
If George is either a person who works from home or has a home office setup, then George is a commuter and is not a person who works from home.

∀x (WorkIn(x, office) → Commuter(x))
∀x (WorkIn(x, office) ⊕ WorkFrom(x, home))
∀x (WorkFrom(x, home) → Have(x, relaxedSchedule))
Commuter(george) ⊕ Have(george, homeOffice)
(WorkFrom(george, home) ⊕ Have(george, homeOffice)) → ¬WorkFrom(george, home) ∧ Commuter(george)


If George is not a person who works from home and a person who works in the office, then George is neither a commuter nor a person who has a relaxed schedule.

¬(WorkFrom(george, home) ∧ WorkIn(george, office)) → ¬(Commuter(george) ∨ Have(george, relaxedSchedule))


If George is either a person who has a home office setup and a person who works in the office, or neither a person who has a home office setup nor a person who works in the office, then George is either a person who works from home or a person who has a relaxed schedule.

¬(Have(george, homeOffice) ⊕ WorkIn(george, office)) → (WorkFrom(george, home) ⊕ Have(george, relaxedSchedule))


Jason Kramer is an American music supervisor.
Some American radio personalities are also music supervisors.
Anyone who hosts a show on a public radio station is a radio personality.
Joe Rogan is a radio personality.
Jason Kramer hosted a show on a public radio station.

MusicSupervisor(jasonKramer) ∧ American(jasonKramer)
∃x ∃y (American(x) ∧ MusicSupervisor(x) ∧ RadioPersonality(x) ∧ (¬(x=y)) ∧ American(y) ∧ MusicSupervisor(y) ∧ RadioPersonality(y))
∀x ∀y((HostShowOn(x, y) ∧ PublicRadioStation(x)) → RadioPersonality(x))
RadioPersonality(joeRogan)
∃x(HostShowOn(jasonKramer, x) ∧ PublicRadioStation(x))


Hermès bags are not made in Italy.
All Birkin bags are Hermès bags.
All Ferraris are made in Italy.
All cars that carry a Ferrari V12 engine are Ferraris.
All cars that are made in Maranello carry a Ferrari V12 engine.
A Lamborghini SUV is not both a Ferrari and made in Maranello.
A Kelly bag is a Hermès bag, or it is a car that carries a Ferrari V12 engine.

∀x ((Bag(x) ∧ Hermès(x)) → ¬MadeIn(x, italy))
∀x ((Bag(x) ∧ Birkin(x)) → Hermès(x))
∀x (Ferrari(x) → MadeIn(x, italy))
∀x ((Car(x) ∧ Carry(x, ferrariV12Engine)) → Ferrrari(x))
∀x ((Car(x) ∧ MadeIn(x, maranello)) → Carry(x, ferrariV12Engine))
¬(Ferrari(lamborghiniSUV) ∧ MadeIn(lamborghiniSUV, maranello))
(Bag(kelly) ∧ Hermès(kelly)) ∨ (Bag(kelly) ∧ Car(kelly) ∧ Carry(kelly, ferrariV12Engine))


A Kelly bag is a Birkin bag made in Maranello.

Bag(kelly) ∧ MadeIn(kelly, maranello) ∧ Birkin(kelly)


A Kelly bag is not both made in Maranello and a Birkin bag.

Bag(kelly) ∧ ¬(MadeIn(kelly, maranello) ∧ Birkin(kelly))


If someone lives in a place named Galicia, then they live in either Spain or Poland.
Spain is in Europe.
Poland is in Europe.
Rochelle lives in Europe.
Dominique does not live in Europe.
Alfonso lives in a place named Galicia.

∀x (∃y (LiveIn(x, y) ∧ Place(y) ∧ Named(y, galicia)) → LiveIn(x, spain) ⊕ LiveIn(x, poland))
∀x (LiveIn(x, spain) → LiveIn(x, europe))
∀x (LiveIn(x, poland) → LiveIn(x, europe))
LiveIn(rochelle, europe)
¬LiveIn(dominique, europe)
∃y (LiveIn(alfonso, y) ∧ Place(y) ∧ Named(y, galicia))

∃y (LiveIn(rochelle, y) ∧ Place(y) ∧ Named(y, galicia))


Ralph Hammerthaler was born in Wasserburg am Inn.
Wasserburg am Inn is in Germany.
Germany is in Europe.
Ralph Hammerthaler is a German writer.
Ralph Hammerthaler was born in 1965.

BornIn(ralphHammerthaler, wasserburgamInn)
LocatedIn(wasserbAmInn, germany)
LocatedIn(germany, europe)
Writer(ralphHammerthaler) ∧ German(ralphHammerthaler)
BornIn1965(ralphHammerthaler)

Ralph Hammerthaler is a writer born in Asia.

Writer(ralphHammerthaler) ∧ BornIn(ralphHammerthaler, asia)


All books written by Neil Gaiman have sold more than one thousand copies.
Some books that have won Hugo Awards are written by Neil Gaiman.
Tomas has read all books written by Tolkien.
Either Tomas has read Hamlet, or it has sold more than one thousand copies.
Hamlet has either sold more than one thousand copies or it is written by Neil Gaiman.

∀x ((Book(x) ∧ WrittenBy(x, neilGaiman)) → ∃y (MoreThan(y, num1000) ∧ SoldCopies(x, y)))
∃x ∃y (Book(x) ∧ Win(x, hugoAward) ∧ WrittenBy(x, neilGaiman) ∧ (¬(x=y)) ∧ Book(y) ∧ Win(y, hugoAward) ∧ WrittenBy(y, neilGaiman))
∀x ((Book(x) ∧ WrittenBy(x, tolkien)) → ReadBy(x, tomas))
ReadBy(hamlet, tomas) ⊕ (∃y (MoreThan(y, num1000) ∧ SoldCopies(hamlet, y)))
∃y (MoreThan(y, num1000) ∧ SoldCopies(hamlet, y)) ⊕ WrittenBy(hamlet, neilGaiman)


Hamlet has won a Hugo Award and is written by Tolkien.

Win(hamlet, hugoAward) ∧ WrittenBy(hamlet, tolkien)


If Hamlet has either won a Hugo Award and is written by Tolkien, or neither has won a Hugo Award nor is written by Tolkien, then Hamlet has neither won a Hugo Award nor is written by Neil Gaiman.

¬(Win(hamlet, hugoAward) ⊕ WrittenBy(hamlet, tolkien)) → (¬Win(hamlet, hugoAward) ∧ (¬WrittenBy(hamlet, neilGaiman)))


Grass is not food
All meadows are grass.
All edible things are food.
All fruits are edible.
All lemons are fruit.
Grapes are not both edible and lemons.
Bananas are grasses or fruits.

∀x (Grass(x) → ¬Food(x))
∀x (Meadow(x) → Grass(x))
∀x (Edible(x) → Food(x))
∀x (Fruit(x) → Edible(x))
∀x (Lemon(x) → Fruit(x))
¬(Edible(grape) ∧ Lemon(grape))
Grass(banana) ∨ Fruit(banana)


Bananas are not both a lemon and a meadow.


The Golden State Warriors are a team from San Francisco.
The Golden State Warriors won the NBA finals.
All teams attending the NBA finals have won many games.
Boston Celtics are a team that lost the NBA finals.
If a team wins the NBA finals, then they will have more income.
If a team wins or loses at the NBA finals, then they are attending the finals.

Team(goldenStateWarriors) ∧ From(goldenStateWarriors, sanFrancisco)
Won(goldenStateWarriors, nbaFinals)
∀x ((Team(x) ∧ Attending(x, nbaFinals)) → WonManyGames(x))
Team(bostonCeltics) ∧ Lost(bostonCeltics, nbaFinals)
∀x ((Team(x) ∧ Won(x, nbaFinals)) → MoreIncome(x))
∀x ((Won(x, nbaFinals) ∨ Lost(x, nbaFinals)) → Attending(x, nbaFinals))

The Boston Celtics are from San Francisco.


The Boston Celtics have more than 30 years of experience.


The Golden State Warriors will have more income from gate receipts.


Maya would only play the violin if her fingers could never be injured.
Volleyball players can injure their ankles, fingers, or shoulder.
Maya is a volleyball player.

Play(maya, violin) → ¬CanInjure(maya, fingers)
∀x (VolleyballPlayer(x) → (CanInjure(ankles) ∧ CanInjure(fingers) ∧ CanInjure(shoulder)))
VolleyballPlayer(maya)


All devices belonging to the company are connected to Google Home.
All devices with the company logo belong to the company.
Each device either has the company logo or belongs to employees.
All of the devices belonging to employees can be connected to the company's wifi.
All of the devices connected to Google Home are controlled by managers.
All of the devices that connect to the company's wifi are easy to operate.
All of the devices that are easy to operate were produced after a new CTO is appointed.
ModelXX was not produced after a new CTO was appointed.

∀x ((Devices(x) ∧ BelongTo(x, company)) → ConnectTo(x, googleHome))
∀x ((Devices(x) ∧ With(x, companyLogo)) → BelongTo(x, company))
∀x ((Devices(x) → (With(x, companyLogo) ⊕ BelongTo(x, employee)))
∀x ((Devices(x) ∧ BelongTo(x, employee)) → CanBeConnectedTo(x, wifi))
∀x ((Devices(x) ∧ ConnectTo(x, googleHome)) → ControlledBy(x, manager))
∀x ((Devices(x) ∧ CanBeConnectedTo(x, wifi)) → EasyToOperate(x))
∀x ((Devices(x) ∧ EasyToOperate(x)) → ProducedAfterNewCTOAppointed(x, company))
Devices(modelXX) ∧ (¬ProducedAfterNewCTOAppointed(modelXX, company))


ModelXX is either produced after a new CTO was appointed or it is controlled by managers.

ProducedAfterNewCTOAppointed(modelXX, theCompany) ⊕ ControlledBy(x, manager)


ModelXX is not with the company logo, and managers do not control it.

¬With(modelXX, companyLogo) ∧ (¬ControlledBy(x, manager))


ModelXX is either with the company logo or controlled by managers.

With(modelXX, companyLogo) ⊕ ControlledBy(x, manager)


All CD players are delicate mechanisms.
No delicate mechanisms are suitable toys for children.

∀x (CDPlayer(x) → DelicateMechanism(x))
∀x (DelicateMechanism(x) → ¬(Toy(x) ∧ SuitableFor(x, children)))

Some CD players are suitable toys for children.

∃x ∃y (CDPlayer(x) ∧ CDPlayer(y) ∧ Toy(x) ∧ Toy(y) ∧ SuitableFor(x, children) ∧ SuitableFor(y, children) ∧ ¬(x=y))


All mammals are living beings.
All elephants are mammals.
All baby elephants are elephants.
Some baby elephants are sleepy.
If Jumbo is a living being, then Jumbo is not both an elephant and a mammal.
If Jumbo is sleepy, then Jumbo is either a baby elephant or a mammal.

∀x (Mammal(x) → LivingBeing(x))
∀x (Elephant(x) → Mammal(x))
∀x (BabyElephant(x) → Elephant(x))
∃x (BabyElephant(x) ∧ Sleepy(x))
LivingBeing(jumbo) → ¬(Elephant(jumbo) ∧ Mammal(jumbo))
Sleepy(jumbo) → BabyElephant(jumbo) ⊕ Mammal(jumbo)


Jumbo is neither sleepy nor a baby elephant.


No planet in the solar system relies on nuclear fusion to generate light.
All stars in the solar system rely on nuclear fusion to generate light.
All celestial bodies in the solar systems that have greater than 0.08 solar masses are stars.
If a celestial body in the solar system has cleared its orbit of other debris and has a nearly spherical shape due to its own gravity, then it is a planet.
If Europa is a celestial body in the solar system that has cleared its orbit of other debris and has a nearly spherical shape due to its own gravity or relies on nuclear fusion to generate light, then Europa is a celestial body in the solar system.

∀x (Planet(x) ∧ In(x, solarSystem) → ¬RelyOnToGenerate(x, nuclearFusion, light))
∀x (Star(x) ∧ In(x, solarSystem) → RelyOnToGenerate(x, nuclearFusion, light))
∀x (CelestialBody(x) ∧ In(x, solarSystem) ∧ GreaterThan(x, solarMass, 0point08) → Star(x))
∀x (CelestialBody(x) ∧ In(x, solarSystem) ∧ (∃y (OrbitOf(y, x) ∧ Clear(x, y) ∧ DebrisFree(y))) ∧ NearlySphericalShape(x, gravity) → Planet(x))
(CelestialBody(europa) ∧ In(europa, solarSystem) ∧ (∃y (OrbitOf(y, x) ∧ Clear(x, y) ∧ DebrisFree(y))) ∧ NearlySphericalShape(europa, gravity)) ∨ RelyOnToGenerate(europa, nuclearFusion, light) → CelestialBody(europa) ∧ In(europa, solarSystem)

Europa is a celestial body in the solar system has cleared its orbit of other debris and has a nearly spherical shape due to its own gravity.


Europa is a celestial body in one of the solar systems that have greater than 0.08 solar masses.

CelestialBody(europa) ∧ In(europa, solarSystem) ∧ GreaterThan(europa, solarMass, 0.08)


Europa is not a celestial body in one of the solar systems that have greater than 0.08 solar masses.

¬(CelestialBody(europa) ∧ In(europa, solarSystem) ∧ GreaterThan(europa, solarMass, 0.08))


If Max listens to music, he either listens to classical music or rap.
All the classical songs that Max listens to are from the 12th century.
If Max is listening to a rap song, then the song is by Kanye West.
All songs by Kanye West are full of lyrics.
All songs that are full of lyrics need to be written with words
It is not true that “As it was” by Harry Styles is classical music that Max listens to and is from the 12th century.
Max listens to "As it was" by Harry Styles.

∀x (MaxListenTo(x) → (ClassicalMusic(x) ⊕ Rap(x)))
∀x (MaxListenTo(x) ∧ ClassicalMusic(x) → Song(x) ∧ From(x, 12thCentury))
∀x (MaxListenTo(x) ∧ Rap(x) → Song(x) ∧ By(x, kanyeWest))
∀x (Song(x) ∧ By(x, kanyeWest) → Song(x) ∧ FullOfLyrics(x))
∀x (Song(x) ∧ FullOfLyrics(x) → NeedToBeWrittenWith(x, words))
¬(ClassicalMusic(asItWasByHarryStyles) ∧ Song(asItWasByHarryStyles) ∧ From(asItWasByHarryStyles, 12thCentury))
MaxListenTo(asItWasByHarryStyles)

“As it was” by Harry Styles needs to be written with words.


"As it was” by Harry Styles is a song from the 12th century.

Song(asItWasByHarryStyles) ∧ From(asItWasByHarryStyles, 12thCentury)


"As it was” by Harry Styles is not both a song from Kanye West and needed to be written with words.

¬(Song(asItWasByHarryStyles) ∧ By(asItWasByHarryStyles, kanyeWest) ∧ NeedToBeWrittenWith(asItWasByHarryStyles, words))


"Your Woman" is a song by the British one-person band White Town.
"Your Woman" song peaked at No. 1 on the UK Singles Chart.
If a song peaked at No.1 at a particular place, it was extremely popular.
"Your Woman" peaked at No. 1 in Iceland, Israel, and Spain.

Produce(whiteTown, yourWoman) ∧ OnePersonBand(whiteTown)
Peak(yourWoman, uKSinglesChart)
∀x ((∃y(Peak(x, y))) → Popular(x))
Peak(yourWoman, iceland) ∧ Peak(yourWoman, israel) ∧ Peak(yourWoman, spain)


White Town did not produce any popular songs.

∀x (Produce(whiteTown, x) → ¬Popular(x))


All functions that represent straight lines on the coordinate plane are linear functions.
No linear functions are non-convex functions.
A function is either a non-convex fuction or a convex function.
All quasi-convex functions are real-valued functions.
All convex functions are quasi-convex functions.
The maximum of quasiconvex functions is a function.
The maximum of quasiconvex functions is a function that represents straight lines on the coordinate plane or it is a convex function or it is not a non-convex function.

∀x (Function(x) ∧ RepresentOn(x, straightLine, coordinatePlane) → LinearFunction(x))
∀x (LinearFunction(x) → ¬NonConvexFunction(x))
∀x (Function(x) → NonConvexFunction(x) ⊕ ConvexFunction(x))
∀x (QuasiConvexFunction(x) → RealValuedFunction(x))
∀x (ConvexFunction(x) → QuasiConvexFunction(x))
Function(maximumOfQuasiConvexFunction)
(Function(maximumOfQuasiConvexFunction) ∧ RepresentOn(maximumOfQuasiConvexFunction, straightLine, coordinatePlane)) ∨ ConvexFunction(maximumOfQuasiConvexFunction) ∨ ¬NonConvexFunction(maximumOfQuasiConvexFunction)

The maximum of quasiconvex functions is a function that represent straight lines on the coordinate plane.


The maximum of quasiconvex functions is not a real-valued function.


The maximum of quasiconvex functions is a quasi-convex function or it is not a real-valued function.


If two soccer teams score the same number of goals in one UCL final during the regular time, they need to play for the extra time.
If two soccer teams score the same number of goals in one UCL final during both regular and extra time, they need to play the penalty shoot-out.
Real Madrid and Atlético Madrid both scored one goal in the 2016 UCL final during the regular time.
Real Madrid and Atlético Madrid both scored zero goals in the 2016 UCL final during the extra time.

∀w ∀x ∀y ∀z (SoccerTeam(x) ∧ SoccerTeam(y) ∧ NumberOfGoalScored(x, z) ∧ NumberOfGoalScored(y, w) ∧ y=w ∧ During(regularTime) → PlayExtra(x, y))
∀x ∀y (SoccerTeam(x) ∧ SoccerTeam(y) ∧ SameScore(x, y) ∧ During(regularTime) ∧ During(extraTime) → PlayPenalty(x, y))
SoccerTeam(realMadrid) ∧ SoccerTeam(atleticoMadrid) ∧ SameScore(realMadrid, atleticoMadrid) ∧ During(regularTime)
SoccerTeam(realMadrid) ∧ SoccerTeam(atleticoMadrid) ∧ SameScore(realMadrid, atleticoMadrid) ∧ During(extraTime)

Real Madrid and Atlético Madrid needed to play a penalty shoot-out in the 2016 UCL final.


Real Madrid and Atlético Madrid did not need to play a penalty shoot-out in the 2016 UCL final.


System 7 is a UK-based electronic dance music band.
Steve Hillage and Miquette Giraudy formed System 7.
Steve Hillage and Miquette Giraudy are former members of the band Gong.
Electric dance music bands are bands.
System 7 has released several club singles.
Club singles are not singles.

BasedIn(system7, uk) ∧ ElectronicDanceMusicBand(system7)
Form(stevehillage, system7) ∧ Form(miquettegiraudy, system7)
FormerMemberOf(stevehillage, gong) ∧ FormerMemberOf(miquettegiraudy, gong)
∀x (ElectronicDanceMusicBand(x) → Band(x))
∃x (ClubSingle(x) ∧ Release(system7, x))
∀x (ClubSingle(x) → ¬Single(x))

System 7 was formed by former members of Gong.

∃x (Form(x, system7) ∧ FormerMemberOf(x, gong))


A summarization model is always faithful if it uses content from the input documents.
Extractive models are summarization models.
An extractive model can only use content from the input documents.

∀x (Model(x) ∧ Summarization(x) ∧ OnlyUseInputDocument(x) → Faithful(x))
∀x (Model(x) ∧ Extractive(x) → Model(x) ∧ Summarization(x))
∀x (Model(x) ∧ Extractive(x) → OnlyUseInputDocument(x))

∀x (Model(x) ∧ Extractive(x) → Faithful(x))


Extractive models are not always faithful.

∃x (Model(x) ∧ Extractive(x) ∧ ¬Faithful(x))


If Robin's friends practice coding questions, then they are not studying to go to medical school to become a doctor.
If Robin's friends want to work in the software engineering industry, then they practice coding questions.
If Robin's friends enjoy healthcare fields and want to help people with medical issues, then they are studying to go to medical school to become a doctor.
If Robin's friends grew up with parents who worked as doctors, then they enjoy healthcare fields and want to help people with medical issues.
If Robin's friends study hard, then they grew up with parents who worked as doctors.
Mark is Robin's friend.
If Mark neither enjoys healthcare fields and wants to help people with medical issues nor grew up with parents who worked as doctors, then Mark is either a person who studies hard or grew up with parents who worked as doctors.

∀x (RobinsFriends(x) ∧ Practice(x, codingQuestion) → ¬StudyingToGoToToBecome(x, medicalSchool, doctor))
∀x (RobinsFriends(x) ∧ WantToWorkIn(x, softwareEngineeringIndustry) → PracticeCodingQuestions(x))
∀x (RobinsFriends(x) ∧ Enjoy(x, healthcareField) ∧ WantToHelp(x, peopleWithMedicalIssue) → StudyingToGoToToBecome(x, medicalSchool, doctor))
∀x (RobinsFriends(x) ∧ (∃y ∃z (¬(y=z) ∧ GrowUpWith(x, y) ∧ GrowUpWith(x, z) ∧ ParentOf(y, x) ∧ ParentOf(z, x) ∧ Doctor(y) ∧ Doctor(z))) → EnjoyHealthcareFields(x) ∧ WantToHelp(x, peopleWithMedicalIssue))
∀x (RobinsFriends(x) ∧ StudyHard(x) → ∃y ∃z (¬(y=z) ∧ GrowUpWith(x, y) ∧ GrowUpWith(x, z) ∧ ParentOf(y, x) ∧ ParentOf(z, x) ∧ Doctor(y) ∧ Doctor(z)))
RobinsFriends(mark)
¬((Enjoy(x, healthcareField) ∧ WantToHelp(mark, peopleWithMedicalIssues)) ∧ ¬(∃y ∃z (¬(y=z) ∧ GrowUpWith(x, y) ∧ GrowUpWith(x, z) ∧ ParentOf(y, x) ∧ ParentOf(z, x) ∧ Doctor(y) ∧ Doctor(z))) → StudyHard(mark) ∨ (∃y ∃z (¬(y=z) ∧ GrowUpWith(x, y) ∧ GrowUpWith(x, z) ∧ ParentOf(y, x) ∧ ParentOf(z, x) ∧ Doctor(y) ∧ Doctor(z)))

Mark is Robin's friend and he is a person who studies hard.


Mark is Robin's friend and he practices coding questions and wants to work in the software engineering industry.

RobinsFriends(mark) ∧ Practice(mark, codingQuestion) ∧ WantToWorkIn(mark, softwareEngineeringIndustry)


Mark is Robin's friend and he neither practices coding questions nor works to work in the software engineering industry.

RobinsFriends(mark) ∧ ¬(Practice(mark, codingQuestion) ∨ WantToWorkIn(mark, softwareEngineeringIndustry))


People who work at Jess's company and go to the spa frequently are not people who are miserly and need to save a large portion of their income.
People who work at Jess's company are either miserly and need to save a large portion of their income, or frivolously spend a lot of money.
If people who work at Jess's company frivolously spend a lot of money when they go out, then they value quality manufacturing and luxury items.
If people who work at Jess's company value quality manufacturing and luxury items, then they enjoy shopping for materialistic items in their free time.
Thomas works at Jess's company.
If Thomas is not miserly and needs to save a large portion of his income, then Thomas does not value quality manufacturing and luxury items.
Thomas values quality manufacturing and luxury items or he is not miserly.

∀x (WorkAt(x, jesssCompany) ∧ GoToSpafrequently(x) → ¬(Miserly(x) ∧ NeedToSave(x, aLargePortionOfIncome)))
∀x (WorkAt(x, jesssCompany) → Miserly(x) ∧ NeedToSave(x, aLargePortionOfIncome)⊕ SpendFrivolously(x, aLotOfMoney))
∀x (WorkAt(x, jesssCompany) ∧ SpendFrivolously(x, aLotOfMoney) → Value(x, qualityManufacturing) ∧ Value(x, luxuryItem))
∀x (WorkAt(x, jesssCompany) ∧ Value(x, qualityManufacturing) ∧ Value(x, luxuryItem) → Enjoy(x, shopping, materialisticItem))
WorkAt(thomas, jesssCompany)
¬(Miserly(thomas) ∧ NeedToSave(thomas, aLargePortionOfIncome)) → ¬((Value(thomas, qualityManufacturing) ∧ Value(thomas, luxuryItem)))
(Value(thomas, qualityManufacturing) ∧ Value(thomas, luxuryItem)) ∨ ¬(Miserly(x) ∧ NeedToSave(x, aLargePortionOfIncome))

Thomas frivolously spends a lot of money.


Thomas either enjoys shopping for materialistic items in his free time or, if he does not, then he goes to the spa frequently.

Enjoy(thomas, shopping, materialisticItem) ⊕ GoToSpaFrequently(thomas)


If Thomas either enjoys shopping for materialistic items in his free time or, if he does not, then he goes to the spa frequently, then Thomas neither values quality manufacturing and luxury items nor goes to the spa frequently.

Enjoy(thomas, shopping, materialisticItem) ⊕ GoToSpaFrequently(thomas) → ¬((Value(x, qualityManufacturing) ∧ Value(x, luxuryItem)) ∨ GoToSpaFrequently(thomas))


The indie pop band Phoenix has released six albums.
Phoenix's album "Wolfgang Amadeus Phoenix" sold over 500,000 copies.
A certified gold album or single is one which sold over half a million copies.
"1901" is a single from Phoenix's album "Wolfgang Amadeus Phoenix."
Over 400,000 copies of "1901" have been sold.

AlbumsReleased(phoenix, 6)
Album(wolfgangamadeusphoenix) ∧ IsAlbumOf(wolfgangamadeusphoenix, phoenix) ∧ SoldOver(wolfgangamadeusphoenix, 500,000)
∀x ((Album(x) ∨ Single(x)) ∧ SoldOver(x, 500,000) → CertifiedGold(x))
Single(1901) ∧ From(1901, wolfgangamadeusphoenix) ∧ By(1901, phoenix)
SoldOver(l1901, 400,000)

The album "Wolfgang Amadeus Phoenix" is a certified gold album.


The single "1901" is a certified gold single.


Peter Parker is either a superhero or a civilian.
The Hulk is a destroyer.
The Hulk wakes up when he is angry.
If the Hulk wakes up, then he will break a bridge.
Thor is a god.
Thor will break a bridge when he is happy.
A god is not a destroyer.
Peter Parker wears a uniform when he is a superhero.
Peter Parker is not a civilian if a destroyer is breaking a bridge.
If Thor is happy, the Hulk is angry.

Superhero(peterParker) ⊕ Civilian(peterParker)
Destroyer(theHulk)
Angry(theHulk) → WakesUp(theHulk)
WakesUp(theHulk) → Breaks(theHulk, bridge)
God(thor)
Happy(thor) → Breaks(thor, bridge)
∀x (God(x) → ¬Destroyer(x))
Superhero(peter) → Wears(peter, uniform)
∀x ((Destroyer(x) ∧ Breaks(x,bridge)) → ¬Civilian(peter))
Happy(thor) → Angry(theHulk)

If the Hulk does not wake up, then Thor is not happy.


If Thor is happy, then Peter Parker wears a uniform.

Happy(thor) → Wears(peterParker, uniform)


If Thor is not happy, then no bridge will be broken.


Diethylcarbamazine is a medication discovered in the year 1947.
Diethylcarbamazine can be used to treat river blindness.
The only preferred treatment for river blindness is ivermectin.
Diethylcarbamazine is not ivermectin.

Medication(diethylcarbamazine) ∧ DiscoversIn(diethylcarbamazine, yr1947)
Treats(diethylcarbamazine, riverBlindness)
PreferredTreatmentFor(riverBlindness, ivermectin)
¬(Is(diethylcarbamazine, ivermectin))

Diethylcarbamazine is not preferred for the treatment of river blindness.


Diethylcarbamazine was often used to treat river blindness.


Diethylcarbamazine is used in the treatment of filariasis.


All prime numbers are natural numbers.
All integers are real numbers.
All real numbers are complex numbers.
One is a prime number or a natural number or both.
If one is not a complex number, then one is a prime number and an integer.

∀x (PrimeNumber(x) → NaturalNumber(x))
∀x (Integer(x) → RealNumber(x))
∀x (RealNumber(x) → ComplexNumber(x))
PrimeNumber(one) ∨ NaturalNumber(one)
¬ComplexNumber(one) → (PrimeNumber(one) ∧ Integer(one))


One is a prime number and a natural number.


One is either a prime number or a natural number.


If some diseases require a medical diagnosis, then lab tests or imaging is required.
All rare diseases require a medical diagnosis.
If a disease is mild, then no lab tests or imaging is required.
All blood cancers are rare diseases.
All types of leukemia are diseases and blood cancers.
Bladder cancer is a disease and is blood cancer or Leukemia.

∀x (Disease(x) ∧ Require(x, medicalDiagnosis) → RequiredFor(labTest, x) ∨ RequiredFor(imaging, x))
∀x (RareDisease(x) → Require(x, medicalDiagnosis))
∀x (Disease(x) ∧ Mild(x) → ¬(RequiredFor(labTest, x) ∨ RequiredFor(imaging, x)))
∀x (BloodCancer(x) → RareDiseases(x))
∀x (Disease(x) ∧ Leukemia(x) → BloodCancer(x))
Disease(bladderCancer) ∧ (BloodCancer(bladderCancer) ∨ Leukemia(bladderCancer))


Bladder cancer is either a rare disease or a mild disease.


There are no elements with atomic number between 61-63 that are not scarce in China.
Non-rare earth elements are not scarce in China.
All elements are either non-rare earth elements or rare earth elements.
All rare earth elements can be used for industry.
All rare earth elements are essential for exploring future directions of electronics.
Lithium is either a non-rare earth element and essential for exploring future directions of electronics, or is not a non-rare earth element and is not essential for exploring future directions of electronics.

∀x ((Element(x) ∧ ∃y(Between(y, num61, num63) ∧ AtomicNumber(x, y))) → ScarceIn(x, china))
∀x (¬RareEarthElement(x) → ¬ScarceIn(x, china))
∀x (¬RareEarthElement(x) ⊕ RareEarthElement(x))
∀x (RareEarthElement(x) → UsedIn(x, industry))
∀x (RareEarthElement(x) → EssentialFor(x, electronics))
¬(¬RareEarthElement(lithium) ⊕ EssentialFor(lithium, electronics))


Lithium is an element with atomic number between 61-63 and is used for batteries.

Element(x) ∧ ∃y(Between(y, num61, num63) ∧ AtomicNumber(x, y)) ∧ UsedFor(lithium, batteries)


If Lithium is not essential for exploring future directions of electronics or an element with atomic number between 61-63, then Lithium is not a non-rare earth element or usable in industry.

¬(EssentialFor(lithium, electronics) ⊕ (∃y(Between(y, num61, num63) ∧ AtomicNumber(lithium, y)))) → ¬(¬RareEarthMetals(lithium) ∨ UsedIn(lithium, industry))


If people don't often clean their homes, then they do not have tidy houses.
If people don't prioritize cleaning, then they do not often clean their homes.
If people hire a maid or cleaning service, then they have tidy houses.
If people don't care about cleanliness, then they do not prioritize cleaning.
Either Jack does hire a maid or cleaning service and does not often clean his home, or he does not hire a maid or cleaning service nor often clean his home.

∀x (¬CleanOften(x, home) → ¬Have(x, tidyHouse))
∀x (¬Prioritize(x, cleaning) → ¬CleanOften(x, home))
∀x (Hire(x, maid) ∨ Hire(x, cleaningService) → Have(x, tidyHouse))
∀x (¬CareAbout(x, cleanliness) → ¬Prioritize(x, cleaning))
¬(Hire(x, maid) ∨ Hire(x, cleaningService)) ⊕ ¬CleanOften(jack, home))


Jack neither lives in the suburbs nor is too busy to clean.

¬(¬CareAbout(jack, cleanliness) ∨ ¬CleanOften(jack, home)


Jack is overburdened and lives in the suburbs.

¬Prioritize(jack, cleaning) ∨ ¬CareAbout(jack, cleanliness)


The bottle not falling is either standing upright or toppled over.
The bottle not falling is not standing upright.

¬Falling(bottle) → (Upright(bottle) ⊕ ToppledOver(bottle))
¬Falling(bottle) → ¬Upright(bottle)


Everyone who chooses what they want to do with their time has flexible schedules.
Everyone with a lot of free time chooses what they want to do with their time.
People either have a lot of free time or they invest in a career in which they are willing to spend the rest of their lives.
If people invest in a career in which they are willing to spend the rest of their lives, then they are hardworking individuals with high ambitions and goals for the future.
If people are hardworking individuals with high ambitions and goals for the future, then they are not short sighted.
John is not either a hardworking individual with high ambitions and goals for the future or has a flexible schedule.

∀x (ChooseWhatToDoWith(x, time) → FlexibleSchedule(x))
∀x (Have(x, lotsOfFreetime) → ChooseWhatToDoWith(x, time))
∀x (Have(x, lotsOfFreetime) ⊕ (∃y (InvestIn(x, y) ∧ Career(y) ∧ WillingToSpendIn(restOfLife, y))))
∀x (∃y (InvestIn(x, y) ∧ Career(y) ∧ WillingToSpendIn(restOfLife, y)) → Hardworking(x))
∀x (Hardworking(x) ∧ HaveFor(x, highAmbition, future) ∧ HaveFor(x, goal, future) → ¬ShortSighted(x))
¬((Hardworking(john) ∧ HaveFor(john, highAmbition, future) ∧ HaveFor(john, goal, future)) ⊕ FlexibleSchedule(john))


John chooses what he want to do with his time.


John is either a hardworking individual with high ambitions and goals for the future or is short sighted.

(Hardworking(john) ∧ HaveFor(john, highAmbition, future) ∧ HaveFor(john, goal, future)) ⊕ ShortSighted(john)


Ableton has an office in Germany.
Ableton has an office in the USA.
USA and Germany are different countries.
Any company that has offices in different countries is a multinational company.
Ableton makes music software.

OfficeIn(ableton, germany)
OfficeIn(ableton, unitedStates)
¬SameCountry(germany, unitedStates)
∀x ∀y ∀z (OfficeIn(x, y) ∧ OfficeIn(x, z) ∧ (¬SameCountry(y, z)) → MultinationalCompany(x))
MakesMusicSoftware(ableton)


Ableton does not have an office in Germany.


Those who can fly over a vast distance glide in the air.
Flightless birds cannot fly over a vast distance.
Penguins are flightless birds.
Nonflying birds in Antarctica are penguins.
Fido is a penguin, or flies over a vast distance.

∀x (FlyOver(x, vastDistance) → GlideInAir(x))
∀x (Flightless(x) ∧ Bird(x) → ¬FlyOver(x, vastDistance))
∀x (Penguin(x) → Flightless(x) ∧ Bird(x))
∀x (NonFlying(x) ∧ Bird(x) ∧ In(x, antarctica) → Penguin(x))
Penguin(fido) ∨ FlyOver(fido, vastDistance)


Fido is not a nonflying bird in Antarctica, and he cannot glid in the air.

¬(NonFlying(fido) ∧ Bird(fido) ∧ In(fido, antarctica)) ∧ ¬GlideInAir(fido)


If Fido either can fly over a vast distance or cannot fly over a vast distance, then Fido is a nonflying bird in Antartica.

(FlyOver(fido, vastDistance) ⊕ ¬FlyOver(fido, vastDistance)) → (NonFlying(fido) ∧ Bird(fido) ∧ In(fido, antarctica))


All members of the university faculty are professors.
All principal investigators are members of the university faculty.
No professor is also an undergraduate student.
Anyone pursuing a bachelor's degree is an undergraduate student.
Leon is not pursuing a bachelor's degree, and he is not a principal investigator.
If Leon is not pursuing a bachelor's degree, then he is a professor.

∀x (MemberOf(x, universityFaculty) → Professor(x))
∀x (PrincipalInvestigator(x) → MemberOf(x, universityFaculty))
∀x (Professor(x) → ¬UndergraduateStudent(x))
∀x (Pursuing(x, bachelor) → UndergraduateStudent(x))
¬(Pursuing(leon, bachelor) ⊕ PrincipalInvestigator(leon))
¬Pursuing(leon, bachelor) → Professor(leon)


Leon is neither an undergraduate student nor a principal investigator.


If leon is not a principal investigator, then Leon is an undergraduate student.