Foundations Of Existential Quantifiers Quiz

1) Which symbol represents "there exists"?

∀

∃

→

¬

The symbol for "there exists" is the existential quantifier, which is denoted by ∃. This symbol is used in logic to express that there is at least one element in the domain that satisfies a given predicate. Therefore, the correct choice is ∃.

Explanation

The symbol for "there exists" is the existential quantifier, which is denoted by ∃. This symbol is used in logic to express that there is at least one element in the domain that satisfies a given predicate. Therefore, the correct choice is ∃.

2) The statement ∃ x P(x) is true when:

P holds for all x

P holds for at least one x

P holds for no x

P is false for all x

An existential statement ∃ x P(x) asserts that there exists at least one x in the domain for which the predicate P(x) is true. Thus, the statement is true if and only if there is at least one x that makes P(x) true. If no such x exists, the statement is false.

Explanation

An existential statement ∃ x P(x) asserts that there exists at least one x in the domain for which the predicate P(x) is true. Thus, the statement is true if and only if there is at least one x that makes P(x) true. If no such x exists, the statement is false.

3) Translate: "Some cats are black."

∀ x(Cat(x) → Black(x))

∃ x(Cat(x) ∧ Black(x))

∀ x(Cat(x) ∧ Black(x))

∃ x(Cat(x) → Black(x))

The phrase "some cats are black" means that there exists at least one cat that is black. In logical terms, this is translated using the existential quantifier ∃ with a conjunction ∧ to ensure that the same x is both a cat and black. The other options are incorrect: ∀ x(Cat(x) → Black(x)) means all cats are black, ∀ x(Cat(x) ∧ Black(x)) means everything is a black cat, and ∃ x(Cat(x) → Black(x)) is not a standard translation for existential claims.

Explanation

The phrase "some cats are black" means that there exists at least one cat that is black. In logical terms, this is translated using the existential quantifier ∃ with a conjunction ∧ to ensure that the same x is both a cat and black. The other options are incorrect: ∀ x(Cat(x) → Black(x)) means all cats are black, ∀ x(Cat(x) ∧ Black(x)) means everything is a black cat, and ∃ x(Cat(x) → Black(x)) is not a standard translation for existential claims.

4) To prove ∃ x P(x), you must:

Find one x where P holds

Show P holds for every x

Assume P holds for all x

Prove P is always false

Proving an existential statement ∃ x P(x) requires demonstrating that there is at least one element in the domain for which P(x) is true. This is typically done by identifying a specific witness—a particular x that satisfies P(x). Showing P holds for every x proves a universal statement, not an existential one.

Explanation

Proving an existential statement ∃ x P(x) requires demonstrating that there is at least one element in the domain for which P(x) is true. This is typically done by identifying a specific witness—a particular x that satisfies P(x). Showing P holds for every x proves a universal statement, not an existential one.

5) The negation of ∃ x P(x) is:

∀ x P(x)

∃ x ¬P(x)

∀ x ¬P(x)

¬∃ x P(x)

The negation of an existential statement is equivalent to a universal statement. By De Morgan's law for quantifiers, ¬∃ x P(x) is logically equivalent to ∀ x ¬P(x). This means that if it is not true that there exists an x with P(x), then for every x, P(x) is false.

Explanation

The negation of an existential statement is equivalent to a universal statement. By De Morgan's law for quantifiers, ¬∃ x P(x) is logically equivalent to ∀ x ¬P(x). This means that if it is not true that there exists an x with P(x), then for every x, P(x) is false.

6) Some people have no siblings:

∃x (Person(x) ∧ ¬∃y (Sibling(y,x) ∧ y ≠ x))

∀x (Person(x) → ¬∃y Sibling(y,x))

∃x ¬Person(x)

∃x ∃y (Person(x) ∧ Sibling(y,x))

"Some people have no siblings" means: There exists a person x such that no one is x's sibling. The clause y ≠ x is included for explicitness—while the sibling relation should be irreflexive by definition (no one is their own sibling), including y ≠ x makes the logic clear and defensive against unusual interpretations. Option B claims all people have no siblings (universal). C is about non-people. D says someone has a sibling, the opposite claim. The correct form uses negated existential within the scope of the outer existential.

Explanation

"Some people have no siblings" means: There exists a person x such that no one is x's sibling. The clause y ≠ x is included for explicitness—while the sibling relation should be irreflexive by definition (no one is their own sibling), including y ≠ x makes the logic clear and defensive against unusual interpretations. Option B claims all people have no siblings (universal). C is about non-people. D says someone has a sibling, the opposite claim. The correct form uses negated existential within the scope of the outer existential.

7) Given ∃ x(P(x) ∧ Q(x)), we can conclude:

∃ x P(x)

P(a) for a specific a

∀ x P(x)

Nothing

From ∃ x(P(x) ∧ Q(x)), we know that there exists some x for which both P(x) and Q(x) are true. Therefore, we can conclude that there exists an x for which P(x) is true, i.e., ∃ x P(x). Similarly, we can conclude ∃ x Q(x). However, we cannot conclude P(a) for a specific a without knowing which x, and we cannot conclude ∀ x P(x).

Explanation

From ∃ x(P(x) ∧ Q(x)), we know that there exists some x for which both P(x) and Q(x) are true. Therefore, we can conclude that there exists an x for which P(x) is true, i.e., ∃ x P(x). Similarly, we can conclude ∃ x Q(x). However, we cannot conclude P(a) for a specific a without knowing which x, and we cannot conclude ∀ x P(x).

8) If ∃ x P(x) is false then ∀ x P(x) is also false:

True

False

If ∃x P(x) is false in a non-empty domain, then P(x) is false for all elements, making ∀x P(x) false. This follows from the equivalence ¬∃x P(x) ≡ ∀x ¬P(x), which holds in standard first-order logic with non-empty domains

Explanation

If ∃x P(x) is false in a non-empty domain, then P(x) is false for all elements, making ∀x P(x) false. This follows from the equivalence ¬∃x P(x) ≡ ∀x ¬P(x), which holds in standard first-order logic with non-empty domains

9) Translate ∃ x(Cat(x) ∧ ¬Black(x)) to English:

All cats are black

Some cats are not black

No cats are black

Some black cats exist

The logical statement ∃ x(Cat(x) ∧ ¬Black(x)) means there exists an x such that x is a cat and x is not black. This translates directly to "some cats are not black." The other options do not match: "All cats are black" would be ∀ x(Cat(x) → Black(x)), "No cats are black" would be ¬∃ x(Cat(x) ∧ Black(x)), and "Some black cats exist" would be ∃ x(Cat(x) ∧ Black(x)).

Explanation

The logical statement ∃ x(Cat(x) ∧ ¬Black(x)) means there exists an x such that x is a cat and x is not black. This translates directly to "some cats are not black." The other options do not match: "All cats are black" would be ∀ x(Cat(x) → Black(x)), "No cats are black" would be ¬∃ x(Cat(x) ∧ Black(x)), and "Some black cats exist" would be ∃ x(Cat(x) ∧ Black(x)).

10) From P(c) we can infer:

∀ x P(x)

∃ x P(x)

P(c) is false

Nothing

If P(c) is true for a specific constant c, then there exists at least one x (namely, c) for which P(x) is true. This allows us to infer the existential statement ∃ x P(x) through existential generalization. We cannot infer ∀ x P(x) from a single example, and P(c) is true by assumption.

Explanation

If P(c) is true for a specific constant c, then there exists at least one x (namely, c) for which P(x) is true. This allows us to infer the existential statement ∃ x P(x) through existential generalization. We cannot infer ∀ x P(x) from a single example, and P(c) is true by assumption.

11) A prime number greater than 2 exists:

∃ x(Prime(x) ∧ x > 2)

∀ x(Prime(x) → x > 2)

∃ x(Prime(x) → x > 2)

∀ x(Prime(x) ∧ x > 2)

The statement asserts the existence of a prime number that is greater than 2. This is correctly translated as ∃ x(Prime(x) ∧ x > 2), where Prime(x) means x is prime, and x > 2 means x is greater than 2. The conjunction ∧ ensures that x is both prime and greater than 2. The other options are incorrect: ∀ x(Prime(x) → x > 2) means all primes are greater than 2, which is false since 2 is prime; ∃ x(Prime(x) → x > 2) is not a standard existential form; and ∀ x(Prime(x) ∧ x > 2) means everything is a prime greater than 2, which is false.

Explanation

The statement asserts the existence of a prime number that is greater than 2. This is correctly translated as ∃ x(Prime(x) ∧ x > 2), where Prime(x) means x is prime, and x > 2 means x is greater than 2. The conjunction ∧ ensures that x is both prime and greater than 2. The other options are incorrect: ∀ x(Prime(x) → x > 2) means all primes are greater than 2, which is false since 2 is prime; ∃ x(Prime(x) → x > 2) is not a standard existential form; and ∀ x(Prime(x) ∧ x > 2) means everything is a prime greater than 2, which is false.

12) Witness for ∃ x(x^2 = 9):

X = 0

X = 3

X = -3

Both B and C

A witness for ∃ x(x^2 = 9) is any x that satisfies the equation x^2 = 9. Both x = 3 and x = -3 satisfy this because 3^2 = 9 and (-3)^2 = 9. Therefore, both are correct witnesses. x = 0 does not satisfy the equation since 0^2 = 0 ≠ 9.

Explanation

A witness for ∃ x(x^2 = 9) is any x that satisfies the equation x^2 = 9. Both x = 3 and x = -3 satisfy this because 3^2 = 9 and (-3)^2 = 9. Therefore, both are correct witnesses. x = 0 does not satisfy the equation since 0^2 = 0 ≠ 9.

13) Some students are not prepared:

∀ x(Student(x) → Prepared(x))

∃ x(Student(x) ∧ ¬Prepared(x))

∀ x(Student(x) ∧ ¬Prepared(x))

∃ x(Student(x) → ¬Prepared(x))

"Some students are not prepared" means that there exists at least one student who is not prepared. This is translated as ∃ x(Student(x) ∧ ¬Prepared(x)), where Student(x) means x is a student, and Prepared(x) means x is prepared. The conjunction ∧ with negation ¬Prepared(x) ensures that x is a student and not prepared. The other options are incorrect: ∀ x(Student(x) → Prepared(x)) means all students are prepared; ∀ x(Student(x) ∧ ¬Prepared(x)) means everything is an unprepared student; and ∃ x(Student(x) → ¬Prepared(x)) is not a standard translation.

Explanation

"Some students are not prepared" means that there exists at least one student who is not prepared. This is translated as ∃ x(Student(x) ∧ ¬Prepared(x)), where Student(x) means x is a student, and Prepared(x) means x is prepared. The conjunction ∧ with negation ¬Prepared(x) ensures that x is a student and not prepared. The other options are incorrect: ∀ x(Student(x) → Prepared(x)) means all students are prepared; ∀ x(Student(x) ∧ ¬Prepared(x)) means everything is an unprepared student; and ∃ x(Student(x) → ¬Prepared(x)) is not a standard translation.

14) Given ∃ x¬P(x) we can infer:

¬∃ x P(x)

∀ x¬P(x)

¬∀ x P(x)

∃ x P(x)

∃ x¬P(x) means that there exists an x for which P(x) is false. This is equivalent to saying that it is not true that P(x) is true for all x, which is ¬∀ x P(x). Therefore, from ∃ x¬P(x), we can infer ¬∀ x P(x). Note that ¬∃ x P(x) is equivalent to ∀ x¬P(x), which is stronger and not directly inferable from ∃ x¬P(x) alone.

Explanation

∃ x¬P(x) means that there exists an x for which P(x) is false. This is equivalent to saying that it is not true that P(x) is true for all x, which is ¬∀ x P(x). Therefore, from ∃ x¬P(x), we can infer ¬∀ x P(x). Note that ¬∃ x P(x) is equivalent to ∀ x¬P(x), which is stronger and not directly inferable from ∃ x¬P(x) alone.

15) Witness for ∃ x(x is even ∧ x is prime):

X = 1

X = 2

X = 3

X = 9

A witness for ∃ x(x is even ∧ x is prime) must be a number that is both even and prime. The only even prime number is 2, as all other even numbers are divisible by 2 and hence not prime. Therefore, x = 2 is the correct witness. x = 1 is not prime, x = 3 is prime but not even, and x = 9 is neither prime nor even.

Explanation

A witness for ∃ x(x is even ∧ x is prime) must be a number that is both even and prime. The only even prime number is 2, as all other even numbers are divisible by 2 and hence not prime. Therefore, x = 2 is the correct witness. x = 1 is not prime, x = 3 is prime but not even, and x = 9 is neither prime nor even.

Foundations of Existential Quantifiers Quiz