lecture 6 AI Flashcards
What is a quantifier in logic?
A quantifier is a symbol or word used to express the quantity of specimens in a statement.
Name the two primary types of quantifiers.
Universal quantifier and existential quantifier.
What symbol represents the universal quantifier?
The universal quantifier is represented by the symbol ‘∀’.
What symbol represents the existential quantifier?
The existential quantifier is represented by the symbol ‘∃’.
True or False: The statement ‘∀x P(x)’ means ‘For all x, P(x) is true’.
True.
True or False: The statement ‘∃x P(x)’ means ‘There exists at least one x such that P(x) is true’.
True.
Fill in the blank: The universal quantifier asserts that a property holds for _____ elements in a domain.
all.
Fill in the blank: The existential quantifier asserts that there is _____ element in a domain for which a property holds.
at least one.
What is the negation of the statement ‘∀x P(x)’?
The negation is ‘∃x ¬P(x)’.
What is the negation of the statement ‘∃x P(x)’?
The negation is ‘∀x ¬P(x)’.
Which quantifier would you use to express ‘All humans are mortal’?
Universal quantifier (∀).
Which quantifier would you use to express ‘Some humans are philosophers’?
Existential quantifier (∃).
Multiple Choice: Which of the following statements is an example of a universal quantifier? A) ∀x (x > 0) B) ∃y (y < 5) C) ∀z (z = z) D) Both A and C
D) Both A and C.
Multiple Choice: Which of the following statements is an example of an existential quantifier? A) ∀x (x + 1 > x) B) ∃y (y = 0) C) ∀z (z < 10) D) Both A and C
B) ∃y (y = 0).
What does the statement ‘∀x (P(x) → Q(x))’ imply?
For every x, if P(x) is true, then Q(x) is also true.
True or False: The expression ‘∃x (P(x) ∧ Q(x))’ means ‘There exists an x such that both P(x) and Q(x) are true’.
True.
Short Answer: What is the primary distinction between universal and existential quantifiers?
Universal quantifiers assert that a property holds for all elements, while existential quantifiers assert that a property holds for at least one element.
Fill in the blank: The statement ‘∃x (P(x) ∨ Q(x))’ means that there exists an x such that _____ P(x) is true or Q(x) is true.
either.
What is the logical form of ‘Not all cats are black’ using quantifiers?
¬∀x (Cat(x) → Black(x)) or ∃x (Cat(x) ∧ ¬Black(x)).
Multiple Choice: Which statement is logically equivalent to ‘∃x P(x)’? A) ¬∀x ¬P(x) B) ∀x P(x) C) ¬∃x ¬P(x) D) Both A and C
D) Both A and C.
True or False: The statement ‘∀x (P(x) ∧ Q(x))’ is equivalent to ‘∀x P(x) ∧ ∀x Q(x)’.
True.
Fill in the blank: The statement ‘∃x (P(x) ∧ ¬Q(x))’ indicates that there is _____ an x for which P(x) is true and Q(x) is false.
at least one.
What is the relationship between quantifiers and predicates in logical statements?
Quantifiers modify predicates to specify the extent to which the predicates apply to elements in a domain.
