Prompt Title: Library Location Logic.

There is a contradiction in the third logical sentence, "Ron is in the library and Ron is not in the library." This statement is logically impossible, as it asserts that Ron is both in the library and not in the library at the same time, which is a contradiction. Therefore, this sentence is false.

Given the first two sentences, "If Hermione is in the library, then Harry is in the library" and "Hermione is in the library," we can logically conclude that Harry is also in the library, as the first sentence establishes a conditional relationship between Hermione and Harry's location.

The fourth sentence, "Harry is in the library," is consistent with the first two sentences and can be inferred from them.

The fifth sentence, "Harry is not in the library or Hermione is in the library," is also consistent with the first two sentences, as it allows for the possibility that Harry is not in the library if Hermione is.

The sixth sentence, "Ron is in the library or Hermione is in the library," is not entailed by the first two sentences but is consistent with them, as it allows for the possibility that either Ron or Hermione (or both) are in the library.

In summary, the only logical entailment that is true is that Harry is in the library, based on the first two sentences.

Based on the given logical sentences:

• Sentence 6 is consistent with Sentence 2, but it does not entail it.
• Sentence 6 is also consistent with Sentence 3, but it does not entail it either.
• Sentence 1 establishes a conditional relationship between Hermione and Harry's location, but it does not entail that Harry is in the library, so it does not entail Sentence 4.
• Sentence 2 states that Hermione is in the library, which is consistent with Sentence 6 allowing for the possibility of Hermione being in the library, but it does not entail Sentence 5.
• Sentence 1 does entail Sentence 2, since the first sentence establishes that if Hermione is in the library, then Harry is in the library, and the second sentence asserts that Hermione is in the library, therefore, Harry must also be in the library.
• Sentence 5 is consistent with Sentence 6, as it allows for the possibility that Hermione is in the library, but it does not entail it.

Therefore, the only logical entailment that is true is Sentence 1 entails Sentence 2.

The expression A ⊕ B represents the sentence "A or B, but not both." In other words, either A is true or B is true, but not both at the same time.

To determine which of the given options is logically equivalent to A ⊕ B, we can create a truth table for both expressions:

From the truth table, we can see that the expression (A ∨ B) ∧ ¬(A ∧ B) has the same truth values as A ⊕ B. Therefore, (A ∨ B) ∧ ¬(A ∧ B) is logically equivalent to A ⊕ B.

In contrast, the other options are not equivalent to A ⊕ B:

• (A ∨ B) ∧ (A ∧ B) is always false, regardless of the truth values of A and B, so it is not equivalent to A ⊕ B.
• (A ∨ B) ∧ ¬(A ∨ B) is always false, regardless of the truth values of A and B, so it is not equivalent to A ⊕ B.
• (A ∧ B) ∨ ¬(A ∨ B) is logically equivalent to A ∧ ¬B ∨ ¬A ∧ B, which is not equivalent to A ⊕ B, as it allows for the possibility of both A and B being true at the same time.

Therefore, the only option that is logically equivalent to A ⊕ B is (A ∨ B) ∧ ¬(A ∧ B).

The symbol (A ∨ B) ∧ ¬(A ∧ B) stands for a logical expression that represents the sentence "A or B, but not both." It is a form of the exclusive or (XOR) logical connective, where either A or B can be true, but not both at the same time.

The expression (A ∨ B) represents the possibility that either A or B (or both) is true, while the expression ¬(A ∧ B) ensures that A and B cannot both be true at the same time. The negation of (A ∧ B) is equivalent to (¬A ∨ ¬B), which means "not A or not B." Thus, the expression (A ∨ B) ∧ ¬(A ∧ B) is true when either A or B is true, but not both, and false otherwise.

The sentence "If it is raining, then it is cloudy and not sunny" can be translated into propositional logic as "If R, then C and not S," which can be written as R → (C ∧ ¬S).

Therefore, the propositional logic representation of the sentence "If it is raining, then it is cloudy and not sunny" is R → (C ∧ ¬S).

The other options are not equivalent to the given sentence:

• (R → C) ∧ ¬S represents the sentence "If it is raining, then it is cloudy, and it is not sunny," which is not equivalent to the original sentence, as it allows for the possibility of both cloudy and sunny weather when it is not raining.
• R → C → ¬S represents the sentence "If it is raining, then if it is cloudy, it is not sunny," which is not equivalent to the original sentence, as it allows for the possibility of both cloudy and sunny weather when it is not raining.
• R ∧ C ∧ ¬S represents the sentence "It is raining, it is cloudy, and it is not sunny," which is not equivalent to the original sentence, as it does not express a conditional relationship between raining, cloudy, and not sunny weather.
• (C ∨ ¬S) → R represents the sentence "If it is either cloudy or not sunny, then it is raining," which is not equivalent to the original sentence, as it allows for the possibility of raining even when it is not cloudy and not sunny.

The first-order logic translation of the sentence "There is a course that Harry and Hermione are both enrolled in" is:

∃x. Course(x) ∧ Enrolled(Harry, x) ∧ Enrolled(Hermione, x)

This sentence states that there exists a course x such that both Harry and Hermione are enrolled in x. The quantifier ∃x ("there exists x") indicates that we are looking for a specific course that satisfies the conditions in the formula. The predicates Course(x), Enrolled(Harry, x), and Enrolled(Hermione, x) specify the properties of the course x, and Harry and Hermione's enrollment in that course.

The other options are not equivalent to the given sentence:

• ∀x. Enrolled(Harry, x) ∨ Enrolled(Hermione, x) represents the sentence "Harry is enrolled in every course or Hermione is enrolled in every course," which does not capture the idea of both Harry and Hermione being enrolled in the same course.
• ∀x. Enrolled(Harry, x) ∧ ∀y. Enrolled(Hermione, y) represents the sentence "Harry is enrolled in every course and Hermione is enrolled in every course," which does not capture the idea of both Harry and Hermione being enrolled in the same course.
• ∃x. Enrolled(Harry, x) ∨ Enrolled(Hermione, x) is similar to the first option and represents the sentence "Harry is enrolled in at least one course that Hermione is also enrolled in, or vice versa," which does not capture the idea of both Harry and Hermione being enrolled in the same course.
• ∀x. Course(x) ∧ Enrolled(Harry, x) ∧ Enrolled(Hermione, x) represents the sentence "Every course has both Harry and Hermione enrolled," which is a stronger statement than what the original sentence is asserting.