Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Using universal domain: All birds have feathers.
Predicates:
- F(x): x is a birds
- G(x): x have feathers

Problem 2: Symbolize: Some rabbits are faster than all turtles.
Predicates:
- F(x): x is a rabbits
- G(y): y is a turtles
- H(x,y): x is faster than y

Problem 3: Using universal domain: All students attend classes.
Predicates:
- F(x): x is a students
- G(x): x attend classes

Problem 4: Using universal domain: Some cars are red.
Predicates:
- F(x): x is a cars
- G(x): x are red

Problem 5: Neither a nor d is G
Predicates:
- G(a): a has property G
- G(d): d has property G

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ax(F(x)⇒G(x))

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))

Answer: Ax(F(x)⇒G(x))

Answer: Ex(F(x)&G(x))

Neither a nor d is G

Answer: ∼G(a)&∼G(d)


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Problem 2: Symbolize: Some rabbits are faster than all turtles.
Predicates:
- F(x): x is a rabbits
- G(y): y is a turtles
- H(x,y): x is faster than y

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: AxEyF(x,y)

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 2: Using universal domain: Some turtles have hard shells.
Predicates:
- F(x): x is a turtles
- G(x): x have hard shells

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ax(F(x)⇒∼G(x))

Answer: Ex(F(x)&G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Using universal domain: All humans breathe.
Predicates:
- F(x): x is a humans
- G(x): x breathe

Problem 2: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ax(F(x)⇒G(x))

Answer: Ax(F(x)⇒∼G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Using universal domain: All humans breathe.
Predicates:
- F(x): x is a humans
- G(x): x breathe

Problem 2: Symbolize: Some students are older than all teachers.
Predicates:
- F(x): x is a students
- G(y): y is a teachers
- H(x,y): x is older than y

Problem 3: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 4: Symbolize: Some humans are smarter than all animals.
Predicates:
- F(x): x is a humans
- G(y): y is a animals
- H(x,y): x is smarter than y

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ax(F(x)⇒G(x))

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))

Answer: Ax(F(x)⇒∼G(x))

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: d is both F and H
Predicates:
- F(d): d has property F
- H(d): d has property H

Problem 2: Using universal domain: Some rabbits have long ears.
Predicates:
- F(x): x is a rabbits
- G(x): x have long ears

Provide answers in [[answer1;answer2;...]] format exactly as required.

d is both F and H

Answer: F(d)&H(d)

Answer: Ex(F(x)&G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 2: Using universal domain: Some turtles have hard shells.
Predicates:
- F(x): x is a turtles
- G(x): x have hard shells

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ax(F(x)⇒∼G(x))

Answer: Ex(F(x)&G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 2: Using universal domain: Some rabbits run fast.
Predicates:
- F(x): x is a rabbits
- G(x): x run fast

Problem 3: Using universal domain: Some turtles have hard shells.
Predicates:
- F(x): x is a turtles
- G(x): x have hard shells

Problem 4: Either b is G or d is G
Predicates:
- G(b): b has property G
- G(d): d has property G

Problem 5: Using universal domain: All students attend classes.
Predicates:
- F(x): x is a students
- G(x): x attend classes

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ax(F(x)⇒∼G(x))

Answer: Ex(F(x)&G(x))

Answer: Ex(F(x)&G(x))

Either b is G or d is G

Answer: G(b)|G(d)

Answer: Ax(F(x)⇒G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 2: Symbolize: Some students are older than all teachers.
Predicates:
- F(x): x is a students
- G(y): y is a teachers
- H(x,y): x is older than y

Problem 3: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 4: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ax(F(x)⇒∼G(x))

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))

Answer: Ax(F(x)⇒∼G(x))

Answer: Ax(F(x)⇒∼G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Symbolize: Some rabbits are faster than all turtles.
Predicates:
- F(x): x is a rabbits
- G(y): y is a turtles
- H(x,y): x is faster than y

Problem 2: Symbolize: Some students are older than all teachers.
Predicates:
- F(x): x is a students
- G(y): y is a teachers
- H(x,y): x is older than y

Problem 3: Neither b nor c is F
Predicates:
- F(b): b has property F
- F(c): c has property F

Problem 4: Using universal domain: Some cars are red.
Predicates:
- F(x): x is a cars
- G(x): x are red

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))

Neither b nor c is F

Answer: ∼F(b)&∼F(c)

Answer: Ex(F(x)&G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Symbolize: Some humans are smarter than all animals.
Predicates:
- F(x): x is a humans
- G(y): y is a animals
- H(x,y): x is smarter than y

Problem 2: Using universal domain: All prime numbers are even.
Predicates:
- F(x): x is a prime numbers
- G(x): x are even

Problem 3: d is both F and H
Predicates:
- F(d): d has property F
- H(d): d has property H

Problem 4: c is both G and H
Predicates:
- G(c): c has property G
- H(c): c has property H

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))

Answer: Ax(F(x)⇒G(x))

d is both F and H

Answer: F(d)&H(d)

c is both G and H

Answer: G(c)&H(c)


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 2: Neither c nor a is G
Predicates:
- G(a): a has property G
- G(c): c has property G

Problem 3: Using universal domain: Some apples are sweet.
Predicates:
- F(x): x is a apples
- G(x): x are sweet

Problem 4: Using universal domain: Some cars are red.
Predicates:
- F(x): x is a cars
- G(x): x are red

Problem 5: Using universal domain: Some rabbits run fast.
Predicates:
- F(x): x is a rabbits
- G(x): x run fast

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ax(F(x)⇒∼G(x))

Neither c nor a is G

Answer: ∼G(c)&∼G(a)

Answer: Ex(F(x)&G(x))

Answer: Ex(F(x)&G(x))

Answer: Ex(F(x)&G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Symbolize: Some humans are smarter than all animals.
Predicates:
- F(x): x is a humans
- G(y): y is a animals
- H(x,y): x is smarter than y

Problem 2: Using universal domain: All students attend classes.
Predicates:
- F(x): x is a students
- G(x): x attend classes

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))

Answer: Ax(F(x)⇒G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Using universal domain: Some turtles swim slowly.
Predicates:
- F(x): x is a turtles
- G(x): x swim slowly

Problem 2: Using universal domain: All birds fly.
Predicates:
- F(x): x is a birds
- G(x): x fly

Problem 3: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 4: Using universal domain: Some cars are red.
Predicates:
- F(x): x is a cars
- G(x): x are red

Problem 5: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ex(F(x)&G(x))

Using universal domain: All birds fly.

Answer: Ax(F(x)⇒G(x))

Answer: Ax(F(x)⇒∼G(x))

Answer: Ex(F(x)&G(x))

Answer: Ax(F(x)⇒∼G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 2: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ax(F(x)⇒∼G(x))

Answer: Ax(F(x)⇒∼G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Symbolize: Some rabbits are faster than all turtles.
Predicates:
- F(x): x is a rabbits
- G(y): y is a turtles
- H(x,y): x is faster than y

Problem 2: If b is G then d is G
Predicates:
- G(b): b has property G
- G(d): d has property G

Problem 3: Using universal domain: All birds have feathers.
Predicates:
- F(x): x is a birds
- G(x): x have feathers

Problem 4: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))

If b is G then d is G

Answer: G(b)⇒G(d)

Answer: Ax(F(x)⇒G(x))

Answer: Ax(F(x)⇒∼G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Using universal domain: Some turtles have hard shells.
Predicates:
- F(x): x is a turtles
- G(x): x have hard shells

Problem 2: Neither b nor a is G
Predicates:
- G(a): a has property G
- G(b): b has property G

Problem 3: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 4: If b is G then c is G
Predicates:
- G(b): b has property G
- G(c): c has property G

Problem 5: Using universal domain: All humans are mortal.
Predicates:
- F(x): x is a humans
- G(x): x are mortal

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ex(F(x)&G(x))

Neither b nor a is G

Answer: ∼G(b)&∼G(a)

Answer: Ax(F(x)⇒∼G(x))

If b is G then c is G

Answer: G(b)⇒G(c)

Answer: Ax(F(x)⇒G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 2: Either a is G or d is G
Predicates:
- G(a): a has property G
- G(d): d has property G

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ax(F(x)⇒∼G(x))

Either a is G or d is G

Answer: G(a)|G(d)


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: c is both F and G
Predicates:
- F(c): c has property F
- G(c): c has property G

Problem 2: Using universal domain: Some turtles have hard shells.
Predicates:
- F(x): x is a turtles
- G(x): x have hard shells

Problem 3: Using universal domain: Some cars are red.
Predicates:
- F(x): x is a cars
- G(x): x are red

Problem 4: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 5: Symbolize: Some students are older than all teachers.
Predicates:
- F(x): x is a students
- G(y): y is a teachers
- H(x,y): x is older than y

Provide answers in [[answer1;answer2;...]] format exactly as required.

c is both F and G

Answer: F(c)&G(c)

Answer: Ex(F(x)&G(x))

Answer: Ex(F(x)&G(x))

Answer: Ax(F(x)⇒∼G(x))

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 2: Using universal domain: Some turtles have hard shells.
Predicates:
- F(x): x is a turtles
- G(x): x have hard shells

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ax(F(x)⇒∼G(x))

Answer: Ex(F(x)&G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Using universal domain: Some apples are sweet.
Predicates:
- F(x): x is a apples
- G(x): x are sweet

Problem 2: Using universal domain: All birds have feathers.
Predicates:
- F(x): x is a birds
- G(x): x have feathers

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ex(F(x)&G(x))

Answer: Ax(F(x)⇒G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Symbolize: Some students are older than all teachers.
Predicates:
- F(x): x is a students
- G(y): y is a teachers
- H(x,y): x is older than y

Problem 2: Using universal domain: All students attend classes.
Predicates:
- F(x): x is a students
- G(x): x attend classes

Problem 3: Symbolize: Some students are older than all teachers.
Predicates:
- F(x): x is a students
- G(y): y is a teachers
- H(x,y): x is older than y

Problem 4: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Problem 5: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))

Answer: Ax(F(x)⇒G(x))

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))

Answer: AxEyF(x,y)

Answer: AxEyF(x,y)


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Neither b nor a is G
Predicates:
- G(a): a has property G
- G(b): b has property G

Problem 2: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 3: Using universal domain: All students study hard.
Predicates:
- F(x): x is a students
- G(x): x study hard

Problem 4: Symbolize: Some rabbits are faster than all turtles.
Predicates:
- F(x): x is a rabbits
- G(y): y is a turtles
- H(x,y): x is faster than y

Problem 5: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Provide answers in [[answer1;answer2;...]] format exactly as required.

Neither b nor a is G

Answer: ∼G(b)&∼G(a)

Answer: Ax(F(x)⇒∼G(x))

Answer: Ax(F(x)⇒G(x))

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))

Answer: AxEyF(x,y)


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Problem 2: Using universal domain: Some apples are organic.
Predicates:
- F(x): x is a apples
- G(x): x are organic

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: AxEyF(x,y)

Answer: Ex(F(x)&G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 2: Neither d nor b is G
Predicates:
- G(b): b has property G
- G(d): d has property G

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ax(F(x)⇒∼G(x))

Neither d nor b is G

Answer: ∼G(d)&∼G(b)


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Symbolize: Some humans are smarter than all animals.
Predicates:
- F(x): x is a humans
- G(y): y is a animals
- H(x,y): x is smarter than y

Problem 2: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 3: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 4: Symbolize: Some students are older than all teachers.
Predicates:
- F(x): x is a students
- G(y): y is a teachers
- H(x,y): x is older than y

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))

Answer: Ax(F(x)⇒∼G(x))

Answer: Ax(F(x)⇒∼G(x))

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Using universal domain: All humans breathe.
Predicates:
- F(x): x is a humans
- G(x): x breathe

Problem 2: Neither a nor c is G
Predicates:
- G(a): a has property G
- G(c): c has property G

Problem 3: Using universal domain: All birds fly.
Predicates:
- F(x): x is a birds
- G(x): x fly

Problem 4: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ax(F(x)⇒G(x))

Neither a nor c is G

Answer: ∼G(a)&∼G(c)

Using universal domain: All birds fly.

Answer: Ax(F(x)⇒G(x))

Answer: AxEyF(x,y)


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 2: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Problem 3: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Problem 4: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ax(F(x)⇒∼G(x))

Answer: AxEyF(x,y)

Answer: AxEyF(x,y)

Answer: Ax(F(x)⇒∼G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 2: Symbolize: Some students are older than all teachers.
Predicates:
- F(x): x is a students
- G(y): y is a teachers
- H(x,y): x is older than y

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ax(F(x)⇒∼G(x))

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 2: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 3: Using universal domain: All birds have feathers.
Predicates:
- F(x): x is a birds
- G(x): x have feathers

Problem 4: Symbolize: Some students are older than all teachers.
Predicates:
- F(x): x is a students
- G(y): y is a teachers
- H(x,y): x is older than y

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ax(F(x)⇒∼G(x))

Answer: Ax(F(x)⇒∼G(x))

Answer: Ax(F(x)⇒G(x))

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: a is both H and F
Predicates:
- F(a): a has property F
- H(a): a has property H

Problem 2: Using universal domain: Some rabbits run fast.
Predicates:
- F(x): x is a rabbits
- G(x): x run fast

Problem 3: Using universal domain: All birds have feathers.
Predicates:
- F(x): x is a birds
- G(x): x have feathers

Problem 4: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Problem 5: Using universal domain: Some cars have turbo engines.
Predicates:
- F(x): x is a cars
- G(x): x have turbo engines

Provide answers in [[answer1;answer2;...]] format exactly as required.

a is both H and F

Answer: H(a)&F(a)

Answer: Ex(F(x)&G(x))

Answer: Ax(F(x)⇒G(x))

Answer: AxEyF(x,y)

Answer: Ex(F(x)&G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Using universal domain: All birds have feathers.
Predicates:
- F(x): x is a birds
- G(x): x have feathers

Problem 2: Using universal domain: All humans are mortal.
Predicates:
- F(x): x is a humans
- G(x): x are mortal

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ax(F(x)⇒G(x))

Answer: Ax(F(x)⇒G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 2: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 3: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Problem 4: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ax(F(x)⇒∼G(x))

Answer: Ax(F(x)⇒∼G(x))

Answer: AxEyF(x,y)

Answer: Ax(F(x)⇒∼G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: a is both F and H
Predicates:
- F(a): a has property F
- H(a): a has property H

Problem 2: Symbolize: Some students are older than all teachers.
Predicates:
- F(x): x is a students
- G(y): y is a teachers
- H(x,y): x is older than y

Problem 3: Using universal domain: Some rabbits run fast.
Predicates:
- F(x): x is a rabbits
- G(x): x run fast

Problem 4: Symbolize: Some humans are smarter than all animals.
Predicates:
- F(x): x is a humans
- G(y): y is a animals
- H(x,y): x is smarter than y

Problem 5: Using universal domain: All humans are mortal.
Predicates:
- F(x): x is a humans
- G(x): x are mortal

Provide answers in [[answer1;answer2;...]] format exactly as required.

a is both F and H

Answer: F(a)&H(a)

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))

Answer: Ex(F(x)&G(x))

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))

Answer: Ax(F(x)⇒G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Symbolize: Some rabbits are faster than all turtles.
Predicates:
- F(x): x is a rabbits
- G(y): y is a turtles
- H(x,y): x is faster than y

Problem 2: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 3: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Problem 4: Symbolize: Some rabbits are faster than all turtles.
Predicates:
- F(x): x is a rabbits
- G(y): y is a turtles
- H(x,y): x is faster than y

Problem 5: d is both H and G
Predicates:
- G(d): d has property G
- H(d): d has property H

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))

Answer: Ax(F(x)⇒∼G(x))

Answer: AxEyF(x,y)

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))

d is both H and G

Answer: H(d)&G(d)


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Using universal domain: All prime numbers are even.
Predicates:
- F(x): x is a prime numbers
- G(x): x are even

Problem 2: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Problem 3: Using universal domain: Some turtles have hard shells.
Predicates:
- F(x): x is a turtles
- G(x): x have hard shells

Problem 4: Using universal domain: Some turtles swim slowly.
Predicates:
- F(x): x is a turtles
- G(x): x swim slowly

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ax(F(x)⇒G(x))

Answer: AxEyF(x,y)

Answer: Ex(F(x)&G(x))

Answer: Ex(F(x)&G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Symbolize: Some humans are smarter than all animals.
Predicates:
- F(x): x is a humans
- G(y): y is a animals
- H(x,y): x is smarter than y

Problem 2: Using universal domain: All prime numbers are greater than 2.
Predicates:
- F(x): x is a prime numbers
- G(x): x are greater than 2

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))

Answer: Ax(F(x)⇒G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Symbolize: Some humans are smarter than all animals.
Predicates:
- F(x): x is a humans
- G(y): y is a animals
- H(x,y): x is smarter than y

Problem 2: Using universal domain: Some turtles swim slowly.
Predicates:
- F(x): x is a turtles
- G(x): x swim slowly

Problem 3: Symbolize: Some humans are smarter than all animals.
Predicates:
- F(x): x is a humans
- G(y): y is a animals
- H(x,y): x is smarter than y

Problem 4: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))

Answer: Ex(F(x)&G(x))

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))

Answer: AxEyF(x,y)


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Problem 2: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 3: Using universal domain: Some rabbits have long ears.
Predicates:
- F(x): x is a rabbits
- G(x): x have long ears

Problem 4: Using universal domain: All birds have feathers.
Predicates:
- F(x): x is a birds
- G(x): x have feathers

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: AxEyF(x,y)

Answer: Ax(F(x)⇒∼G(x))

Answer: Ex(F(x)&G(x))

Answer: Ax(F(x)⇒G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Either c is G or b is G
Predicates:
- G(b): b has property G
- G(c): c has property G

Problem 2: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Provide answers in [[answer1;answer2;...]] format exactly as required.

Either c is G or b is G

Answer: G(c)|G(b)

Answer: Ax(F(x)⇒∼G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Symbolize: Some rabbits are faster than all turtles.
Predicates:
- F(x): x is a rabbits
- G(y): y is a turtles
- H(x,y): x is faster than y

Problem 2: Symbolize: Some students are older than all teachers.
Predicates:
- F(x): x is a students
- G(y): y is a teachers
- H(x,y): x is older than y

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Using universal domain: Some rabbits have long ears.
Predicates:
- F(x): x is a rabbits
- G(x): x have long ears

Problem 2: Using universal domain: Some turtles swim slowly.
Predicates:
- F(x): x is a turtles
- G(x): x swim slowly

Problem 3: Using universal domain: All prime numbers are greater than 2.
Predicates:
- F(x): x is a prime numbers
- G(x): x are greater than 2

Problem 4: Symbolize: Some students are older than all teachers.
Predicates:
- F(x): x is a students
- G(y): y is a teachers
- H(x,y): x is older than y

Problem 5: If b is G then a is G
Predicates:
- G(a): a has property G
- G(b): b has property G

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ex(F(x)&G(x))

Answer: Ex(F(x)&G(x))

Answer: Ax(F(x)⇒G(x))

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))

If b is G then a is G

Answer: G(b)⇒G(a)


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Problem 2: Symbolize: Some rabbits are faster than all turtles.
Predicates:
- F(x): x is a rabbits
- G(y): y is a turtles
- H(x,y): x is faster than y

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: AxEyF(x,y)

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Symbolize: Some rabbits are faster than all turtles.
Predicates:
- F(x): x is a rabbits
- G(y): y is a turtles
- H(x,y): x is faster than y

Problem 2: Neither a nor c is F
Predicates:
- F(a): a has property F
- F(c): c has property F

Problem 3: b is both G and H
Predicates:
- G(b): b has property G
- H(b): b has property H

Problem 4: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 5: Using universal domain: All birds fly.
Predicates:
- F(x): x is a birds
- G(x): x fly

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))

Neither a nor c is F

Answer: ∼F(a)&∼F(c)

b is both G and H

Answer: G(b)&H(b)

Answer: Ax(F(x)⇒∼G(x))

Using universal domain: All birds fly.

Answer: Ax(F(x)⇒G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Symbolize: Some humans are smarter than all animals.
Predicates:
- F(x): x is a humans
- G(y): y is a animals
- H(x,y): x is smarter than y

Problem 2: Symbolize: Some rabbits are faster than all turtles.
Predicates:
- F(x): x is a rabbits
- G(y): y is a turtles
- H(x,y): x is faster than y

Problem 3: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 4: Using universal domain: All birds fly.
Predicates:
- F(x): x is a birds
- G(x): x fly

Problem 5: Using universal domain: Some cars have turbo engines.
Predicates:
- F(x): x is a cars
- G(x): x have turbo engines

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))

Answer: Ax(F(x)⇒∼G(x))

Using universal domain: All birds fly.

Answer: Ax(F(x)⇒G(x))

Answer: Ex(F(x)&G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Problem 2: If c is F then d is F
Predicates:
- F(c): c has property F
- F(d): d has property F

Problem 3: Neither c nor b is F
Predicates:
- F(b): b has property F
- F(c): c has property F

Problem 4: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: AxEyF(x,y)

If c is F then d is F

Answer: F(c)⇒F(d)

Neither c nor b is F

Answer: ∼F(c)&∼F(b)

Answer: AxEyF(x,y)


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 2: Using universal domain: Some cars are red.
Predicates:
- F(x): x is a cars
- G(x): x are red

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ax(F(x)⇒∼G(x))

Answer: Ex(F(x)&G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Symbolize: Some humans are smarter than all animals.
Predicates:
- F(x): x is a humans
- G(y): y is a animals
- H(x,y): x is smarter than y

Problem 2: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 3: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 4: c is both G and H
Predicates:
- G(c): c has property G
- H(c): c has property H

Problem 5: Using universal domain: All students attend classes.
Predicates:
- F(x): x is a students
- G(x): x attend classes

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))

Answer: Ax(F(x)⇒∼G(x))

Answer: Ax(F(x)⇒∼G(x))

c is both G and H

Answer: G(c)&H(c)

Answer: Ax(F(x)⇒G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Using universal domain: Some apples are organic.
Predicates:
- F(x): x is a apples
- G(x): x are organic

Problem 2: b is both H and F
Predicates:
- F(b): b has property F
- H(b): b has property H

Problem 3: Symbolize: Some rabbits are faster than all turtles.
Predicates:
- F(x): x is a rabbits
- G(y): y is a turtles
- H(x,y): x is faster than y

Problem 4: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ex(F(x)&G(x))

b is both H and F

Answer: H(b)&F(b)

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))

Answer: AxEyF(x,y)


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Symbolize: Some rabbits are faster than all turtles.
Predicates:
- F(x): x is a rabbits
- G(y): y is a turtles
- H(x,y): x is faster than y

Problem 2: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Problem 3: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Problem 4: Symbolize: Some rabbits are faster than all turtles.
Predicates:
- F(x): x is a rabbits
- G(y): y is a turtles
- H(x,y): x is faster than y

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))

Answer: AxEyF(x,y)

Answer: AxEyF(x,y)

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Using universal domain: Some apples are organic.
Predicates:
- F(x): x is a apples
- G(x): x are organic

Problem 2: Using universal domain: Some rabbits run fast.
Predicates:
- F(x): x is a rabbits
- G(x): x run fast

Problem 3: Using universal domain: Some turtles swim slowly.
Predicates:
- F(x): x is a turtles
- G(x): x swim slowly

Problem 4: Using universal domain: All students attend classes.
Predicates:
- F(x): x is a students
- G(x): x attend classes

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ex(F(x)&G(x))

Answer: Ex(F(x)&G(x))

Answer: Ex(F(x)&G(x))

Answer: Ax(F(x)⇒G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Problem 2: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Problem 3: Symbolize: Some rabbits are faster than all turtles.
Predicates:
- F(x): x is a rabbits
- G(y): y is a turtles
- H(x,y): x is faster than y

Problem 4: Using universal domain: Some turtles swim slowly.
Predicates:
- F(x): x is a turtles
- G(x): x swim slowly

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: AxEyF(x,y)

Answer: AxEyF(x,y)

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))

Answer: Ex(F(x)&G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: b is both F and G
Predicates:
- F(b): b has property F
- G(b): b has property G

Problem 2: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 3: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 4: Using universal domain: Some rabbits run fast.
Predicates:
- F(x): x is a rabbits
- G(x): x run fast

Problem 5: Either b is F or d is F
Predicates:
- F(b): b has property F
- F(d): d has property F

Provide answers in [[answer1;answer2;...]] format exactly as required.

b is both F and G

Answer: F(b)&G(b)

Answer: Ax(F(x)⇒∼G(x))

Answer: Ax(F(x)⇒∼G(x))

Answer: Ex(F(x)&G(x))

Either b is F or d is F

Answer: F(b)|F(d)


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Using universal domain: Some cars have turbo engines.
Predicates:
- F(x): x is a cars
- G(x): x have turbo engines

Problem 2: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Problem 3: Symbolize: Some rabbits are faster than all turtles.
Predicates:
- F(x): x is a rabbits
- G(y): y is a turtles
- H(x,y): x is faster than y

Problem 4: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 5: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ex(F(x)&G(x))

Answer: AxEyF(x,y)

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))

Answer: Ax(F(x)⇒∼G(x))

Answer: Ax(F(x)⇒∼G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Symbolize: Some rabbits are faster than all turtles.
Predicates:
- F(x): x is a rabbits
- G(y): y is a turtles
- H(x,y): x is faster than y

Problem 2: Using universal domain: Some apples are sweet.
Predicates:
- F(x): x is a apples
- G(x): x are sweet

Problem 3: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Problem 4: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 5: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))

Answer: Ex(F(x)&G(x))

Answer: AxEyF(x,y)

Answer: Ax(F(x)⇒∼G(x))

Answer: Ax(F(x)⇒∼G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 2: Symbolize: Some humans are smarter than all animals.
Predicates:
- F(x): x is a humans
- G(y): y is a animals
- H(x,y): x is smarter than y

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ax(F(x)⇒∼G(x))

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Using universal domain: Some apples are organic.
Predicates:
- F(x): x is a apples
- G(x): x are organic

Problem 2: Using universal domain: All birds have feathers.
Predicates:
- F(x): x is a birds
- G(x): x have feathers

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ex(F(x)&G(x))

Answer: Ax(F(x)⇒G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Problem 2: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: AxEyF(x,y)

Answer: AxEyF(x,y)


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Using universal domain: All prime numbers are even.
Predicates:
- F(x): x is a prime numbers
- G(x): x are even

Problem 2: Neither a nor b is G
Predicates:
- G(a): a has property G
- G(b): b has property G

Problem 3: Using universal domain: All birds fly.
Predicates:
- F(x): x is a birds
- G(x): x fly

Problem 4: Using universal domain: All humans are mortal.
Predicates:
- F(x): x is a humans
- G(x): x are mortal

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ax(F(x)⇒G(x))

Neither a nor b is G

Answer: ∼G(a)&∼G(b)

Using universal domain: All birds fly.

Answer: Ax(F(x)⇒G(x))

Answer: Ax(F(x)⇒G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Problem 2: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: AxEyF(x,y)

Answer: AxEyF(x,y)


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Symbolize: Some rabbits are faster than all turtles.
Predicates:
- F(x): x is a rabbits
- G(y): y is a turtles
- H(x,y): x is faster than y

Problem 2: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Problem 3: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Problem 4: Symbolize: Some students are older than all teachers.
Predicates:
- F(x): x is a students
- G(y): y is a teachers
- H(x,y): x is older than y

Problem 5: Using universal domain: All humans are mortal.
Predicates:
- F(x): x is a humans
- G(x): x are mortal

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))

Answer: AxEyF(x,y)

Answer: AxEyF(x,y)

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))

Answer: Ax(F(x)⇒G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Problem 2: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: AxEyF(x,y)

Answer: Ax(F(x)⇒∼G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Using universal domain: Some apples are organic.
Predicates:
- F(x): x is a apples
- G(x): x are organic

Problem 2: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 3: c is both G and F
Predicates:
- F(c): c has property F
- G(c): c has property G

Problem 4: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ex(F(x)&G(x))

Answer: Ax(F(x)⇒∼G(x))

c is both G and F

Answer: G(c)&F(c)

Answer: Ax(F(x)⇒∼G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Using universal domain: Some apples are sweet.
Predicates:
- F(x): x is a apples
- G(x): x are sweet

Problem 2: Using universal domain: All prime numbers are even.
Predicates:
- F(x): x is a prime numbers
- G(x): x are even

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ex(F(x)&G(x))

Answer: Ax(F(x)⇒G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Problem 2: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: AxEyF(x,y)

Answer: Ax(F(x)⇒∼G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 2: Symbolize: Some students are older than all teachers.
Predicates:
- F(x): x is a students
- G(y): y is a teachers
- H(x,y): x is older than y

Problem 3: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 4: Using universal domain: Some rabbits run fast.
Predicates:
- F(x): x is a rabbits
- G(x): x run fast

Problem 5: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ax(F(x)⇒∼G(x))

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))

Answer: Ax(F(x)⇒∼G(x))

Answer: Ex(F(x)&G(x))

Answer: AxEyF(x,y)


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: If d is F then a is F
Predicates:
- F(a): a has property F
- F(d): d has property F

Problem 2: Using universal domain: All prime numbers are even.
Predicates:
- F(x): x is a prime numbers
- G(x): x are even

Provide answers in [[answer1;answer2;...]] format exactly as required.

If d is F then a is F

Answer: F(d)⇒F(a)

Answer: Ax(F(x)⇒G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Problem 2: Symbolize: Some students are older than all teachers.
Predicates:
- F(x): x is a students
- G(y): y is a teachers
- H(x,y): x is older than y

Problem 3: Symbolize: Some students are older than all teachers.
Predicates:
- F(x): x is a students
- G(y): y is a teachers
- H(x,y): x is older than y

Problem 4: Using universal domain: All prime numbers are even.
Predicates:
- F(x): x is a prime numbers
- G(x): x are even

Problem 5: Symbolize: Some rabbits are faster than all turtles.
Predicates:
- F(x): x is a rabbits
- G(y): y is a turtles
- H(x,y): x is faster than y

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: AxEyF(x,y)

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))

Answer: Ax(F(x)⇒G(x))

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Using universal domain: All birds fly.
Predicates:
- F(x): x is a birds
- G(x): x fly

Problem 2: Symbolize: Some rabbits are faster than all turtles.
Predicates:
- F(x): x is a rabbits
- G(y): y is a turtles
- H(x,y): x is faster than y

Problem 3: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 4: Symbolize: Some humans are smarter than all animals.
Predicates:
- F(x): x is a humans
- G(y): y is a animals
- H(x,y): x is smarter than y

Provide answers in [[answer1;answer2;...]] format exactly as required.

Using universal domain: All birds fly.

Answer: Ax(F(x)⇒G(x))

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))

Answer: Ax(F(x)⇒∼G(x))

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Using universal domain: All birds fly.
Predicates:
- F(x): x is a birds
- G(x): x fly

Problem 2: Neither d nor b is G
Predicates:
- G(b): b has property G
- G(d): d has property G

Problem 3: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 4: Symbolize: Some rabbits are faster than all turtles.
Predicates:
- F(x): x is a rabbits
- G(y): y is a turtles
- H(x,y): x is faster than y

Provide answers in [[answer1;answer2;...]] format exactly as required.

Using universal domain: All birds fly.

Answer: Ax(F(x)⇒G(x))

Neither d nor b is G

Answer: ∼G(d)&∼G(b)

Answer: Ax(F(x)⇒∼G(x))

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Using universal domain: Some apples are sweet.
Predicates:
- F(x): x is a apples
- G(x): x are sweet

Problem 2: Using universal domain: Some apples are organic.
Predicates:
- F(x): x is a apples
- G(x): x are organic

Problem 3: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 4: Symbolize: Some humans are smarter than all animals.
Predicates:
- F(x): x is a humans
- G(y): y is a animals
- H(x,y): x is smarter than y

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ex(F(x)&G(x))

Answer: Ex(F(x)&G(x))

Answer: Ax(F(x)⇒∼G(x))

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 2: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ax(F(x)⇒∼G(x))

Answer: AxEyF(x,y)


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Symbolize: Some rabbits are faster than all turtles.
Predicates:
- F(x): x is a rabbits
- G(y): y is a turtles
- H(x,y): x is faster than y

Problem 2: Neither c nor d is F
Predicates:
- F(c): c has property F
- F(d): d has property F

Problem 3: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Problem 4: Either d is G or a is G
Predicates:
- G(a): a has property G
- G(d): d has property G

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))

Neither c nor d is F

Answer: ∼F(c)&∼F(d)

Answer: AxEyF(x,y)

Either d is G or a is G

Answer: G(d)|G(a)


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Using universal domain: All humans are mortal.
Predicates:
- F(x): x is a humans
- G(x): x are mortal

Problem 2: Using universal domain: All humans breathe.
Predicates:
- F(x): x is a humans
- G(x): x breathe

Problem 3: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 4: Using universal domain: All humans are mortal.
Predicates:
- F(x): x is a humans
- G(x): x are mortal

Problem 5: Symbolize: Some humans are smarter than all animals.
Predicates:
- F(x): x is a humans
- G(y): y is a animals
- H(x,y): x is smarter than y

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ax(F(x)⇒G(x))

Answer: Ax(F(x)⇒G(x))

Answer: Ax(F(x)⇒∼G(x))

Answer: Ax(F(x)⇒G(x))

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 2: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 3: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 4: Using universal domain: All birds fly.
Predicates:
- F(x): x is a birds
- G(x): x fly

Problem 5: Either b is F or c is F
Predicates:
- F(b): b has property F
- F(c): c has property F

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ax(F(x)⇒∼G(x))

Answer: Ax(F(x)⇒∼G(x))

Answer: Ax(F(x)⇒∼G(x))

Using universal domain: All birds fly.

Answer: Ax(F(x)⇒G(x))

Either b is F or c is F

Answer: F(b)|F(c)


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: If c is F then a is F
Predicates:
- F(a): a has property F
- F(c): c has property F

Problem 2: Using universal domain: Some rabbits run fast.
Predicates:
- F(x): x is a rabbits
- G(x): x run fast

Problem 3: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Problem 4: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 5: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Provide answers in [[answer1;answer2;...]] format exactly as required.

If c is F then a is F

Answer: F(c)⇒F(a)

Answer: Ex(F(x)&G(x))

Answer: AxEyF(x,y)

Answer: Ax(F(x)⇒∼G(x))

Answer: Ax(F(x)⇒∼G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 2: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Problem 3: c is both G and F
Predicates:
- F(c): c has property F
- G(c): c has property G

Problem 4: Using universal domain: All prime numbers are even.
Predicates:
- F(x): x is a prime numbers
- G(x): x are even

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ax(F(x)⇒∼G(x))

Answer: AxEyF(x,y)

c is both G and F

Answer: G(c)&F(c)

Answer: Ax(F(x)⇒G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Using universal domain: All birds have feathers.
Predicates:
- F(x): x is a birds
- G(x): x have feathers

Problem 2: b is both H and F
Predicates:
- F(b): b has property F
- H(b): b has property H

Problem 3: Using universal domain: Some rabbits have long ears.
Predicates:
- F(x): x is a rabbits
- G(x): x have long ears

Problem 4: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Problem 5: Using universal domain: Some apples are sweet.
Predicates:
- F(x): x is a apples
- G(x): x are sweet

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ax(F(x)⇒G(x))

b is both H and F

Answer: H(b)&F(b)

Answer: Ex(F(x)&G(x))

Answer: AxEyF(x,y)

Answer: Ex(F(x)&G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Neither b nor a is F
Predicates:
- F(a): a has property F
- F(b): b has property F

Problem 2: Using universal domain: All students study hard.
Predicates:
- F(x): x is a students
- G(x): x study hard

Provide answers in [[answer1;answer2;...]] format exactly as required.

Neither b nor a is F

Answer: ∼F(b)&∼F(a)

Answer: Ax(F(x)⇒G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Using universal domain: Some rabbits have long ears.
Predicates:
- F(x): x is a rabbits
- G(x): x have long ears

Problem 2: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ex(F(x)&G(x))

Answer: AxEyF(x,y)


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Problem 2: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Problem 3: Symbolize: Some humans are smarter than all animals.
Predicates:
- F(x): x is a humans
- G(y): y is a animals
- H(x,y): x is smarter than y

Problem 4: Using universal domain: Some turtles swim slowly.
Predicates:
- F(x): x is a turtles
- G(x): x swim slowly

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: AxEyF(x,y)

Answer: AxEyF(x,y)

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))

Answer: Ex(F(x)&G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Symbolize: Some humans are smarter than all animals.
Predicates:
- F(x): x is a humans
- G(y): y is a animals
- H(x,y): x is smarter than y

Problem 2: Symbolize: Some rabbits are faster than all turtles.
Predicates:
- F(x): x is a rabbits
- G(y): y is a turtles
- H(x,y): x is faster than y

Problem 3: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Problem 4: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 5: Either b is G or c is G
Predicates:
- G(b): b has property G
- G(c): c has property G

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))

Answer: AxEyF(x,y)

Answer: Ax(F(x)⇒∼G(x))

Either b is G or c is G

Answer: G(b)|G(c)


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Symbolize: Some rabbits are faster than all turtles.
Predicates:
- F(x): x is a rabbits
- G(y): y is a turtles
- H(x,y): x is faster than y

Problem 2: c is both F and G
Predicates:
- F(c): c has property F
- G(c): c has property G

Problem 3: Using universal domain: All students study hard.
Predicates:
- F(x): x is a students
- G(x): x study hard

Problem 4: If a is G then d is G
Predicates:
- G(a): a has property G
- G(d): d has property G

Problem 5: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))

c is both F and G

Answer: F(c)&G(c)

Answer: Ax(F(x)⇒G(x))

If a is G then d is G

Answer: G(a)⇒G(d)

Answer: Ax(F(x)⇒∼G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: If b is F then c is F
Predicates:
- F(b): b has property F
- F(c): c has property F

Problem 2: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Problem 3: Using universal domain: All students attend classes.
Predicates:
- F(x): x is a students
- G(x): x attend classes

Problem 4: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Problem 5: Symbolize: Some rabbits are faster than all turtles.
Predicates:
- F(x): x is a rabbits
- G(y): y is a turtles
- H(x,y): x is faster than y

Provide answers in [[answer1;answer2;...]] format exactly as required.

If b is F then c is F

Answer: F(b)⇒F(c)

Answer: AxEyF(x,y)

Answer: Ax(F(x)⇒G(x))

Answer: AxEyF(x,y)

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Using universal domain: Some rabbits run fast.
Predicates:
- F(x): x is a rabbits
- G(x): x run fast

Problem 2: d is both F and G
Predicates:
- F(d): d has property F
- G(d): d has property G

Problem 3: Using universal domain: All prime numbers are even.
Predicates:
- F(x): x is a prime numbers
- G(x): x are even

Problem 4: Symbolize: Some students are older than all teachers.
Predicates:
- F(x): x is a students
- G(y): y is a teachers
- H(x,y): x is older than y

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ex(F(x)&G(x))

d is both F and G

Answer: F(d)&G(d)

Answer: Ax(F(x)⇒G(x))

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 2: Symbolize: Some students are older than all teachers.
Predicates:
- F(x): x is a students
- G(y): y is a teachers
- H(x,y): x is older than y

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ax(F(x)⇒∼G(x))

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Symbolize: Some rabbits are faster than all turtles.
Predicates:
- F(x): x is a rabbits
- G(y): y is a turtles
- H(x,y): x is faster than y

Problem 2: Using universal domain: All humans breathe.
Predicates:
- F(x): x is a humans
- G(x): x breathe

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))

Answer: Ax(F(x)⇒G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Symbolize: Some students are older than all teachers.
Predicates:
- F(x): x is a students
- G(y): y is a teachers
- H(x,y): x is older than y

Problem 2: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Problem 3: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Problem 4: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))

Answer: AxEyF(x,y)

Answer: AxEyF(x,y)

Answer: AxEyF(x,y)


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Using universal domain: Some turtles have hard shells.
Predicates:
- F(x): x is a turtles
- G(x): x have hard shells

Problem 2: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ex(F(x)&G(x))

Answer: AxEyF(x,y)


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Symbolize: Some students are older than all teachers.
Predicates:
- F(x): x is a students
- G(y): y is a teachers
- H(x,y): x is older than y

Problem 2: Using universal domain: Some turtles have hard shells.
Predicates:
- F(x): x is a turtles
- G(x): x have hard shells

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))

Answer: Ex(F(x)&G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Using universal domain: Some apples are sweet.
Predicates:
- F(x): x is a apples
- G(x): x are sweet

Problem 2: Using universal domain: All humans are mortal.
Predicates:
- F(x): x is a humans
- G(x): x are mortal

Problem 3: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Problem 4: Either a is G or c is G
Predicates:
- G(a): a has property G
- G(c): c has property G

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ex(F(x)&G(x))

Answer: Ax(F(x)⇒G(x))

Answer: AxEyF(x,y)

Either a is G or c is G

Answer: G(a)|G(c)


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Using universal domain: Some turtles have hard shells.
Predicates:
- F(x): x is a turtles
- G(x): x have hard shells

Problem 2: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Problem 3: Using universal domain: Some rabbits have long ears.
Predicates:
- F(x): x is a rabbits
- G(x): x have long ears

Problem 4: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ex(F(x)&G(x))

Answer: AxEyF(x,y)

Answer: Ex(F(x)&G(x))

Answer: Ax(F(x)⇒∼G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Problem 2: Symbolize: Some humans are smarter than all animals.
Predicates:
- F(x): x is a humans
- G(y): y is a animals
- H(x,y): x is smarter than y

Problem 3: If d is F then c is F
Predicates:
- F(c): c has property F
- F(d): d has property F

Problem 4: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: AxEyF(x,y)

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))

If d is F then c is F

Answer: F(d)⇒F(c)

Answer: Ax(F(x)⇒∼G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Using universal domain: Some apples are organic.
Predicates:
- F(x): x is a apples
- G(x): x are organic

Problem 2: Neither c nor d is F
Predicates:
- F(c): c has property F
- F(d): d has property F

Problem 3: Using universal domain: Some rabbits have long ears.
Predicates:
- F(x): x is a rabbits
- G(x): x have long ears

Problem 4: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 5: Either a is G or d is G
Predicates:
- G(a): a has property G
- G(d): d has property G

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ex(F(x)&G(x))

Neither c nor d is F

Answer: ∼F(c)&∼F(d)

Answer: Ex(F(x)&G(x))

Answer: Ax(F(x)⇒∼G(x))

Either a is G or d is G

Answer: G(a)|G(d)


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Symbolize: Some humans are smarter than all animals.
Predicates:
- F(x): x is a humans
- G(y): y is a animals
- H(x,y): x is smarter than y

Problem 2: Using universal domain: All humans breathe.
Predicates:
- F(x): x is a humans
- G(x): x breathe

Problem 3: Using universal domain: Some apples are organic.
Predicates:
- F(x): x is a apples
- G(x): x are organic

Problem 4: Either c is F or a is F
Predicates:
- F(a): a has property F
- F(c): c has property F

Problem 5: If d is G then b is G
Predicates:
- G(b): b has property G
- G(d): d has property G

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))

Answer: Ax(F(x)⇒G(x))

Answer: Ex(F(x)&G(x))

Either c is F or a is F

Answer: F(c)|F(a)

If d is G then b is G

Answer: G(d)⇒G(b)


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Either d is F or a is F
Predicates:
- F(a): a has property F
- F(d): d has property F

Problem 2: Using universal domain: All students study hard.
Predicates:
- F(x): x is a students
- G(x): x study hard

Problem 3: Symbolize: Some students are older than all teachers.
Predicates:
- F(x): x is a students
- G(y): y is a teachers
- H(x,y): x is older than y

Problem 4: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Problem 5: Using universal domain: All birds fly.
Predicates:
- F(x): x is a birds
- G(x): x fly

Provide answers in [[answer1;answer2;...]] format exactly as required.

Either d is F or a is F

Answer: F(d)|F(a)

Answer: Ax(F(x)⇒G(x))

Answer: Ex(F(x)&Ay(G(y)⇒H(x,y)))

Answer: AxEyF(x,y)

Using universal domain: All birds fly.

Answer: Ax(F(x)⇒G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Neither c nor a is G
Predicates:
- G(a): a has property G
- G(c): c has property G

Problem 2: Using universal domain: All humans are mortal.
Predicates:
- F(x): x is a humans
- G(x): x are mortal

Provide answers in [[answer1;answer2;...]] format exactly as required.

Neither c nor a is G

Answer: ∼G(c)&∼G(a)

Answer: Ax(F(x)⇒G(x))


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Using universal domain: Some rabbits have long ears.
Predicates:
- F(x): x is a rabbits
- G(x): x have long ears

Problem 2: No humans can fly. (Using universal domain)
Predicates:
- F(x): x is human
- G(x): x can fly

Problem 3: Using universal domain: Some rabbits have long ears.
Predicates:
- F(x): x is a rabbits
- G(x): x have long ears

Problem 4: Using universal domain: All birds fly.
Predicates:
- F(x): x is a birds
- G(x): x fly

Problem 5: Neither d nor b is G
Predicates:
- G(b): b has property G
- G(d): d has property G

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ex(F(x)&G(x))

Answer: Ax(F(x)⇒∼G(x))

Answer: Ex(F(x)&G(x))

Using universal domain: All birds fly.

Answer: Ax(F(x)⇒G(x))

Neither d nor b is G

Answer: ∼G(d)&∼G(b)


Prompt:
In first-order logic, symbolize the following propositions using the given predicates.
Strictly follow these notation rules:
- Universal Quantifier: Ax (for all x)
- Existential Quantifier: Ex (there exists x)
- Logical Connectives: & (and), | (or), ⇒ (implies), ∼ (not)
- Predicate format: Use capitalized letters with variables (e.g., F(x), G(x,y))
- 0-ary predicates must use constants (e.g., F(a), G(b,c))

Problem 1: Using universal domain: All humans breathe.
Predicates:
- F(x): x is a humans
- G(x): x breathe

Problem 2: Every person has someone they love. (Domain: people)
Predicates:
- F(x,y): x loves y

Provide answers in [[answer1;answer2;...]] format exactly as required.

Answer: Ax(F(x)⇒G(x))

Answer: AxEyF(x,y)