Existential Statements Quiz: Examine Existence Claims in Logic

A single witness satisfying both predicates ensures each existential is true individually. If one element 𝑐 makes both 𝑃(𝑐) and 𝑄(𝑐) true, then that same 𝑐 serves as a witness for both ∃𝑥𝑃(𝑥) and ∃𝑥𝑄(𝑥). A single object verifying both predicates guarantees each existential claim holds.

The integers have no maximum; they extend without bound. For any integer 𝑛, the number 𝑛+1 is always larger.

Because the set Z has no greatest element, the statement

∃𝑥 “x is the largest integer” must be false.

Both 3 and −3 satisfy x²=9 because 3²=9 and (−3)²=9, so each is a valid witness for the existential claim ∃x(x²=9).

In a singleton domain {a}, both ∃xP(x) and ∀xP(x) reduce to checking P(a), so existential and universal quantifiers have the same truth condition.

Equality is reflexive, so every element satisfies x=x; therefore in any nonempty domain, ∃x(x=x) is guaranteed to be true.

∃x(P(x) ∧ Q(x)) is stronger than ∃xP(x) ∧ ∃xQ(x) because the former requires one element satisfying both predicates, whereas the latter may use two different witnesses.

Negating an existential yields a universal with a negated predicate: ¬∃xP(x) ≡ ∀x¬P(x), following De Morgan’s law for quantifiers.

Negating ∀xP(x) ∧ ∀xQ(x) gives ∃x¬P(x) ∨ ∃x¬Q(x) because negation distributes over ∧ and universal quantifiers become existential ones.

If some element a satisfies P(a,y) for all y, then for any specific y=c we still have P(a,c); thus ∃x∀yP(x,y) implies ∀y∃xP(x,y).

The statement ∃x∃y(x≠y) requires at least two distinct elements, so a domain of size 2 is the minimum that can make it true.

∃xP(x) and ∃x¬P(x) can both be true because they may have different witnesses, one satisfying P and another satisfying ¬P.

In any nonempty domain:



If some element satisfies

𝑃, the left disjunct is true.



If no element satisfies

𝑃, then all elements satisfy ¬P, so the right disjunct is true.



Thus, one of the two sides is always true → a tautology.



Only if empty domains were allowed (some advanced logics). Standard logic disallows empty domains, so your answer stands.

The statement ∃xP(x) ∨ ∀x¬P(x) is a tautology in any nonempty domain because if some element satisfies P the left disjunct is true, and if no element satisfies P then all satisfy ¬P, making the right disjunct true.

An existential statement asserts that there exists at least one element in the domain for which the predicate holds. It does not require the predicate to be true for many elements or even for most—just a single valid witness is enough to make the entire existential claim true.

A statement like ∃x (P(x) ∧ Q(x)) demands that one specific element satisfy both conditions simultaneously. It is not enough that one element satisfies P and a different one satisfies Q; the existential claim binds them together through the same variable.

In ordinary language, expressions such as “some number,” “some person,” or “there is at least one” translate directly into the existential quantifier ∃. This form does not require identifying the witness explicitly—just knowing that at least one exists.

To prove an existential statement, it is sufficient to exhibit a single example that satisfies the predicate. Providing a witness immediately establishes the truth of ∃x P(x), because the existence claim becomes grounded in a specific instance.

A statement like ∃x (P(x) ∨ Q(x)) is equivalent to (∃x P(x)) ∨ (∃x Q(x)). This works because finding an x that satisfies either predicate already guarantees that the existential claim is fulfilled—only one path needs to succeed.

In expressions such as ∃x ∃y R(x, y), swapping the quantifiers to ∃y ∃x R(x, y) leaves the meaning unchanged. Both forms simply claim the existence of some x and some y that satisfy the relation, and the independence of the witnesses makes the order irrelevant.

A prime number has exactly two distinct positive divisors. The number 2 meets this definition since its only divisors are 1 and 2. Any even number n > 2 is divisible by 2 and by at least one additional number, meaning it has more than two divisors and cannot be prime.