Quantifiers Fundamentals Logic Quiz – Undergraduate Predicate Logic Practice

The symbol ∀ is the universal quantifier in predicate logic, representing "for all" or "for every." The symbol ∃ (option A) is the existential quantifier meaning "there exists," ∧ (option B) represents logical AND, and → (option D) represents implication. The universal quantifier ∀ is used to make statements about all elements in a domain.

The symbol ∃ is the existential quantifier, which means "there exists" or "for some." The expression ∃x P(x) asserts that at least one element in the domain satisfies the predicate P. It does not require P(x) to be true for all elements, only for at least one element. This is different from the universal quantifier ∀ which would mean P(x) is true for every x.

The symbol ∀ is the universal quantifier, meaning "for all" or "for every." The expression ∀x P(x) means that the predicate P is true for every element in the domain. This is a strong statement that requires P(x) to hold for all possible values of x, not just some values. If even one element in the domain fails to satisfy P, then ∀x P(x) would be false.

The negation of a universal statement "for all x, P(x)" is equivalent to "there exists an x such that not P(x)." If it's not true that P holds for every element, then there must be at least one element for which P does not hold. This is a fundamental equivalence in predicate logic: ¬(∀x P(x)) ≡ ∃x ¬P(x). Option A would mean P is false for all x, which is stronger than what we need.

The negation of an existential statement "there exists an x such that P(x)" is equivalent to "for all x, not P(x)." If there is no x that satisfies P, then all elements in the domain must fail to satisfy P. This is another fundamental equivalence: ¬(∃x P(x)) ≡ ∀x ¬P(x). Option B is actually the negation of the universal statement, not the existential.

In predicate logic, universal statements like "Every student passed" require a conditional (→) rather than a conjunction (∧). The correct translation means "For all x, if x is a student, then x passed." Using conjunction (option C) would incorrectly mean "For all x, x is a student and x passed," which implies everyone in the domain is a student, which isn't what the original statement says. The universal quantifier with conditional captures that among all entities, those who are students also passed.

"Some cats are black" means there exists at least one entity that is both a cat and black. This is correctly captured by the existential quantifier with conjunction. Using a universal quantifier with implication (option B) would mean "All cats are black," which is stronger than the original statement. The existential quantifier with conjunction ensures we're only claiming the existence of at least one black cat, not that all cats are black.

The statement 'Some cats are black' is ∃x(Cat(x) ∧ Black(x)). Its negation is ∀x(Cat(x) → ¬Black(x)), which means 'No cats are black' - for every cat, it is not black. This is the correct negation because if it's not true that some cats are black, then no cats can be black. Option C ('Some cats are not black') would be the negation of 'All cats are black,' not the negation we need here.

"No dogs are reptiles" means for all x, if x is a dog, then x is not a reptile. This is correctly represented by the universal quantifier with conditional. Option C would mean "There exists a dog that is a reptile," which is the opposite of what we want. Option D would mean nothing is either a dog or a reptile, which is much stronger than the original statement.

The statement ∃x ¬P(x) means there exists at least one element x in the domain for which P(x) is false. This is equivalent to saying "Some x makes P(x) false." It does not mean P(x) is false for all x (which would be ∀x ¬P(x)), just that there is at least one counterexample.

The statement "Every human is mortal" is a classic universal conditional statement. It means for all x, if x is human, then x is mortal. Using conjunction would incorrectly claim that everything in the domain is both human and mortal. The conditional form correctly captures that among all entities, those who are human also have the property of being mortal.

The statement claims the existence of at least one number that is both prime and greater than 100. This is correctly represented by the existential quantifier with conjunction. Option B would mean all prime numbers are greater than 100, which is not what the statement says. Option C would be true even if there were no prime numbers greater than 100 (since a conditional with false antecedent is true).

The statement means for each x in the domain, if P(x) holds, then 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 that satisfies P(x). For any x where P(x) is false, the implication is vacuously true regardless of y. This is a crucial distinction in predicate logic - when the existential quantifier is within the scope of the universal quantifier, the witness (y) can depend on the value of x.

The domain of discourse is the set of all possible values that variables can take in a logical expression. Quantifiers like ∀ and ∃ range over this domain, specifying how statements apply to elements within this set. It defines the context for interpreting quantified statements - for example, if the domain is "all humans," then ∀x P(x) means "all humans have property P."

The statement ∃x ∀y loves(x,y) reads as "There exists an x such that for all y, x loves y." This means there is a single person (x) who loves every person (y) in the domain. This is different from ∀y ∃x loves(x,y), which would mean "Everyone is loved by someone" (possibly different people loving different individuals). The order of quantifiers is crucial for the meaning.