Quantifier Order, Domains, And Number Properties Quiz

1) Negation of “Every dog is friendly” is:

Some dog is not friendly

No dog is friendly

Every dog is not friendly

Some dog is friendly

The negation of "All dogs are friendly" is "There exists at least one dog that is not friendly." This is because to disprove a universal statement, we only need one counterexample.

Explanation

The negation of "All dogs are friendly" is "There exists at least one dog that is not friendly." This is because to disprove a universal statement, we only need one counterexample.

2) “∀x ∃y (x > y)” over real numbers is:

True

False

Undefined

Contradiction

For any real number x, we can find a smaller real number y (e.g., y = x-1). Since this is true for all real numbers, the statement is true.

Explanation

For any real number x, we can find a smaller real number y (e.g., y = x-1). Since this is true for all real numbers, the statement is true.

3) “No student failed” translates to:

∀x (Student(x) → ¬Failed(x))

∃x (Student(x) → ¬Failed(x))

∀x (¬Student(x) ∧ Failed(x))

∃x (Student(x) ∧ Failed(x))

"No student failed" means that if someone is a student, they did not fail. This is a universal conditional statement: for all x, if x is a student, then x did not fail.

Explanation

"No student failed" means that if someone is a student, they did not fail. This is a universal conditional statement: for all x, if x is a student, then x did not fail.

4) There is no smallest real number can be represented as:

¬∃x ∀y(x ≤ y)

∃x ∀y(x ≤ y)

∀x ∃y(x ≤ y)

∃x ¬∀y(x ≤ y)

To say there is no smallest real number means it's not the case that there exists a number x that is less than or equal to all numbers y. This is correctly expressed as the negation of "there exists an x such that for all y, x ≤ y." Option C merely states that every number has a number greater than or equal to it, which is true but doesn't address whether a minimum exists. This question helps students understand how to formalize statements with "there is no" constructions using quantifier negation.

Explanation

To say there is no smallest real number means it's not the case that there exists a number x that is less than or equal to all numbers y. This is correctly expressed as the negation of "there exists an x such that for all y, x ≤ y." Option C merely states that every number has a number greater than or equal to it, which is true but doesn't address whether a minimum exists. This question helps students understand how to formalize statements with "there is no" constructions using quantifier negation.

5) “∀x ∈ ℝ (x² ≥ 0)” is an example of a:

Universal statement

Existential statement

Conditional

Contradiction

The statement uses a universal quantifier (∀) over the domain of real numbers, asserting that the property x² ≥ 0 holds for all elements in the domain. Note that this statement is also a tautology (always true), but the question asks for the type based on quantifier structure.

Explanation

The statement uses a universal quantifier (∀) over the domain of real numbers, asserting that the property x² ≥ 0 holds for all elements in the domain. Note that this statement is also a tautology (always true), but the question asks for the type based on quantifier structure.

6) “∃x ∈ ℕ (x² = 9)” is an example of:

Universal statement

Existential statement

Contradiction

Implication

The statement uses an existential quantifier (∃) over the domain of natural numbers, asserting that there exists at least one element in the domain for which the property holds.

Explanation

The statement uses an existential quantifier (∃) over the domain of natural numbers, asserting that there exists at least one element in the domain for which the property holds.

7) “∀x (P(x) ∨ Q(x))” means:

For Every x, at least one of P(x) or Q(x) is true

There exists an x for which both are true

For Every x, neither holds

There exists no x such that P(x)

The universal disjunction states that for every element in the domain, either P(x) is true, Q(x) is true, or both are true.

Explanation

The universal disjunction states that for every element in the domain, either P(x) is true, Q(x) is true, or both are true.

8) “∃x (P(x) ∨ Q(x))” means:

At least one x makes P(x) or Q(x) true

For all x, both are true

No x satisfies either

All x make P(x) false

The existential disjunction states that there is at least one element in the domain for which either P(x) is true, Q(x) is true, or both are true.

Explanation

The existential disjunction states that there is at least one element in the domain for which either P(x) is true, Q(x) is true, or both are true.

9) Which pair of statements are logically equivalent?

¬∀x P(x) and ∃x ¬P(x)

∀x P(x) and ∃x ¬P(x)

∃x P(x) and ∀x ¬P(x)

¬∃x P(x) and ∀x P(x)

These two statements are logically equivalent by the rules of quantifier negation. The negation of a universal statement is equivalent to an existential statement with the negated predicate.

Explanation

These two statements are logically equivalent by the rules of quantifier negation. The negation of a universal statement is equivalent to an existential statement with the negated predicate.

10) The quantifier order in “∀x ∃y” differs from “∃y ∀x” because:

The order affects the meaning

The order ne ⅴ er matters

Both are equivalent

They’re both contradictions

The order of quantifiers significantly affects the meaning of a statement. "∀x ∃y" means for each x there is a y (possibly different for each x), while "∃y ∀x" means there is one y that works for all x.

Explanation

The order of quantifiers significantly affects the meaning of a statement. "∀x ∃y" means for each x there is a y (possibly different for each x), while "∃y ∀x" means there is one y that works for all x.

11) “∀x ∃y (x + y = 0)” over real numbers is:

A) True

B) False

C) Undefined

D) Contradiction

For every real number x, there exists a real number y (specifically y = -x) such that x + y = 0. This statement is true for all real numbers.

Explanation

For every real number x, there exists a real number y (specifically y = -x) such that x + y = 0. This statement is true for all real numbers.

12) “∃x ∀y (x + y = y)” over reals is:

True

False

Undefined

Contradiction

The statement is true when x = 0, because 0 + y = y for all real numbers y. Since there exists at least one x (namely 0) that satisfies the condition for all y, the statement is true.

Explanation

The statement is true when x = 0, because 0 + y = y for all real numbers y. Since there exists at least one x (namely 0) that satisfies the condition for all y, the statement is true.

13) “∃x ∀y (x + y = x)” over reals is:

True

False

Undefined

Contradiction

This statement would require an x such that for all y, x + y = x, which simplifies to y = 0 for all y. Since this is not true (there are real numbers other than 0), the statement is false.

Explanation

This statement would require an x such that for all y, x + y = x, which simplifies to y = 0 for all y. Since this is not true (there are real numbers other than 0), the statement is false.

14) “Every nonzero number has a reciprocal” means:

∀x (x ≠ 0 → ∃y (x·y = 1))

∃x (x ≠ 0 ∧ ∀y (x·y = 1))

∀x ∀y (x·y = 1)

∀x(x ≠ 0 → ∃y(y·x = 1))

The statement means for all x that are not zero, there exists a y such that x·y = 1. The universal quantifier applies to x (with the condition x ≠ 0), and the existential quantifier applies to y. Note that in formal logic, we avoid notations like 1/x in the formalization since this assumes what we're trying to define.

Explanation

The statement means for all x that are not zero, there exists a y such that x·y = 1. The universal quantifier applies to x (with the condition x ≠ 0), and the existential quantifier applies to y. Note that in formal logic, we avoid notations like 1/x in the formalization since this assumes what we're trying to define.

15) ∀x(P(x) → ∃yQ(x,y)) allows that:

The same y works for all x

Different y may be needed for different x

No y exists for any x

All x share the same y

The statement means for each x that satisfies P(x), there exists some y that makes Q(x,y) true. The existential quantifier is inside the scope of the universal quantifier, so the y that works may be different for each x.

Explanation

The statement means for each x that satisfies P(x), there exists some y that makes Q(x,y) true. The existential quantifier is inside the scope of the universal quantifier, so the y that works may be different for each x.