Mastering Predicates and Quantifiers

What is a logical predicate?

• A statement that can be true or false based on variable values.

• A mathematical equation that cannot be evaluated.

• A statement that is always true regardless of variable values.

• A type of logical fallacy used in arguments.

What does the existential quantifier signify?

• There exists no element that satisfies a given property.

• At least two elements satisfy a given property.

• All elements satisfy a given property.

• There exists at least one element that satisfies a given property.

Negate the statement 'For every x, P(x) is true'.

• For some x, P(x) is true.

• There exists an x such that P(x) is true.

• There exists an x such that P(x) is not true.

• For every x, P(x) is false.

Evaluate the logical expression: ∀x (P(x) ∨ Q(x)) when P(1) is true and Q(1) is false.

• Q(x) is true for all x.

• The expression is true for all x.

• P(1) is sufficient to evaluate the expression.

• Cannot determine without more information about P(x) and Q(x) for all x.

What is the difference between universal and existential quantifiers?

• Universal quantifiers apply to some elements; existential quantifiers apply to none.

• Universal quantifiers are used only in logic; existential quantifiers are used in mathematics.

• Universal quantifiers are always true; existential quantifiers are always false.

• Universal quantifiers assert properties for all elements; existential quantifiers assert properties for at least one element.

Negate the statement 'There exists an x such that Q(x) is false'.

• There exists an x such that Q(x) is true.

• For some x, Q(x) is false.

• There is no x such that Q(x) is true.

• For all x, Q(x) is true.

What is the symbol for the universal quantifier?

• ⊥

• ∈

• ∀

• ∃

Negate the statement 'For all x, Q(x) is true'.

• There exists an x such that Q(x) is true.

• For some x, Q(x) is true.

• There exists an x such that Q(x) is not true.

• For all x, Q(x) is false.

Evaluate the logical expression: ∀x (P(x) → Q(x)) when P(3) is true and Q(3) is false.

• true

• P(3) is false

• undefined

• false

What does the statement 'There exists a y such that P(y) is true' imply?

• It implies that at least one instance of y makes P(y) true.

• It means P(y) is false for all y.

• It suggests that P(y) is true for every possible y.

• It indicates that y must be a specific value.