Negating Quantifiers and Translating Quantified Statements Quiz

The negation of "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 doesn't hold. This is the quantifier negation rule: ¬∀x P(x) ≡ ∃x ¬P(x).

The statement ¬∃x ¬P(x) means "It is not true that there exists an x such that not P(x)." By De Morgan's law for quantifiers, this is equivalent to 'For all x, not(not P(x))'. This is equivalent to saying "For all x, P(x) is true." This is another application of the quantifier negation rules: the negation of "there exists an x such that not P(x)" is "for all x, P(x)." This is equivalent to the principle that if something isn't false for any element, then it must be true for all elements.

"Some birds cannot fly" means there exists at least one entity that is both a bird and cannot fly. This is correctly captured by the existential quantifier with conjunction. Option B would mean "All birds can fly," which is the opposite of what we want. The statement specifically identifies some birds that lack the ability to fly, so we need an existential statement about birds with the property of not being able to fly.

The statement means for all integers x, it's not true that x is both even and odd. This is a universal statement that says the property of being both even and odd never holds for any integer. Option B would mean the opposite - that there is at least one integer that is both even and odd. The correct translation uses the universal quantifier to assert that no element in the domain has this contradictory property.

This is the standard interpretation of a universal conditional statement. If P(x) implies Q(x) for all x, then whenever P(x) is true for some x, Q(x) must also be true for that same x. It doesn't mean that P(x) is true for all x, just that whenever P(x) holds, Q(x) must also hold. This is the definition of logical implication in predicate logic.

The existential quantifier ∃ means "there exists" or "there is at least one." This is the standard symbol for expressing "at least one" in predicate logic. The universal quantifier ∀ (option B) means "for all," which is the opposite of what we need. The other symbols are logical connectives, not quantifiers.

A free variable is one that is not bound by a quantifier. In the expression ∀x P(x,y), the variable x is bound by the universal quantifier (it's the variable being quantified over), while y has no quantifier and is therefore free. Free variables represent parameters that can take on different values, while bound variables are "local" to the quantified expression.

The statement is an existential claim that there is at least one value of x for which x² equals 4 (in the real numbers, x=2 and x=-2 both satisfy this). This is the direct meaning of the existential quantifier - it asserts the existence of at least one example. The statement doesn't claim anything about other values of x.

The negation of "there exists an x such that P(x)" means that there is no x for which P(x) is true - i.e., P(x) is false for all x. This is logically equivalent to ∀x ¬P(x). The statement asserts the complete absence of any elements satisfying P(x).

The negation of "for all x, P(x)" means it's not the case that P(x) is true for every x. This is equivalent to saying there exists at least one x where P(x) is false. It does not mean that P(x) is false for all x (which would be stronger), just that there is at least one counterexample to the universal claim.

This is an existential statement requiring a conjunction: there exists an x that is both a number and even. This correctly captures that at least one number is even.

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. This captures the mathematical fact that each nonzero number has its own reciprocal, rather than there being one universal reciprocal for all numbers.

This statement claims there exists an integer that is greater than all integers. Since the integers are unbounded above (there is no greatest integer), this statement is false.

For any integer x, we can find another integer y (such as x+1) that is greater than x. Since this is true for all integers, the statement is true.

The statement means for every integer x, there exists an integer y that is the successor of x (i.e., y = x+1). This requires a universal quantifier followed by an existential quantifier.