Quiz 1: Knowledge¶
CS50's Introduction to Artificial Intelligence with Python
Question 1¶
Which of the following logical entailments is true?
- Sentence 6 entails Sentence 2
- Sentence 1 entails Sentence 4
- Sentence 6 entails Sentence 3
- Sentence 2 entails Sentence 5 ✓
- Sentence 1 entails Sentence 2
- Sentence 5 entails Sentence 6
Why: α ⊨ β means in every world where α is true, β is also true. Evaluate each sentence pair by checking if the truth of the first forces the truth of the second across all models.
Question 2¶
Which of the following is logically equivalent to A ⊕ B? (where ⊕ represents exclusive or)
(A ∨ B) ∧ ¬(A ∧ B)✓(A ∧ B) ∨ ¬(A ∨ B)(A ∨ B) ∧ (A ∧ B)(A ∨ B) ∧ ¬(A ∨ B)
Why: Exclusive or is true when exactly one of A or B is true.
(A ∨ B)ensures at least one is true.¬(A ∧ B)ensures both aren't true simultaneously. The conjunction captures "one or the other, but not both."
Question 3¶
Which of the following is a propositional logic representation of "If it is raining, then it is cloudy and not sunny"?
(R → C) ∧ ¬SR → C → ¬SR ∧ C ∧ ¬SR → (C ∧ ¬S)✓(C ∨ ¬S) → R
Why: The implication
R → (C ∧ ¬S)reads "if R, then (C and not S)." The parentheses matter — both C and ¬S must follow from R together, not separately.
Question 4¶
Which of the following is a first-order logic translation of "There is a course that Harry and Hermione are both enrolled in"?
∃x. Course(x) ∧ Enrolled(Harry, x) ∧ Enrolled(Hermione, x)✓∀x. Course(x) ∧ Enrolled(Harry, x) ∧ Enrolled(Hermione, x)∃x. Enrolled(Harry, x) ∧ ∃y. Enrolled(Hermione, y)∀x. Enrolled(Harry, x) ∧ ∀y. Enrolled(Hermione, y)∃x. Enrolled(Harry, x) ∨ Enrolled(Hermione, x)∀x. Enrolled(Harry, x) ∨ Enrolled(Hermione, x)
Why: "There is a course" → existential quantifier ∃. The same variable x must appear in both Enrolled predicates to ensure it's the same course, not two different ones.