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Consider the first-order logic sentence
φ ≡ ∃s∃t∃u∀v∀w∀x∀y ψ(s, t, u, v, w, x, y)
where ψ(s, t, u, v, w, x, y) is a quantifier-free first-order logic formula using only predicate symbols, and possibly equality, but no function symbols. Suppose φ has a model with a universe containing 7 elements. Which one of the following statements is necessarily true?
φ ≡ ∃s∃t∃u∀v∀w∀x∀y ψ(s, t, u, v, w, x, y)
where ψ(s, t, u, v, w, x, y) is a quantifier-free first-order logic formula using only predicate symbols, and possibly equality, but no function symbols. Suppose φ has a model with a universe containing 7 elements. Which one of the following statements is necessarily true?
- a)There exists at least one model of φ with universe of size less than or equal to 3
- b)There exists no model of φ with universe of size less than or equal to 3
- c)There exists no model of φ with universe size of greater than 7
- d)Every model of φ has a universe of size equal to 7
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
Consider the first-order logic sentenceφ ≡ ∃s∃t&...
Let's interpret the problem this way : ∀ are always True and ∃ are always False for empty sets. So there exists at least one model with universe of size 3 (or less than). Therefore, option (A) is necessarily TRUE.
Most Upvoted Answer
Consider the first-order logic sentenceφ ≡ ∃s∃t&...
Understanding the Statement
The first-order logic sentence φ states that there exist elements s, t, u in the universe such that for all elements v, w, x, and y, the formula ψ(s, t, u, v, w, x, y) holds.
Key Insights
- Quantifier-Free Formula: Since ψ is quantifier-free, it can be satisfied in a certain configuration of elements in the universe.
- Existential and Universal Quantifiers: The existential quantifiers for s, t, and u indicate that we only need to find values for these three variables. The universal quantifiers for v, w, x, and y imply that the formula must hold for every possible combination of these remaining variables.
Reasons Why Option A is True
- Non-uniqueness of Models: The existence of a model with 7 elements does not restrict the number of elements in other possible models.
- Smaller Universes: It is possible to construct models of φ with fewer than 7 elements. For instance, if you can satisfy ψ with just 3 distinct elements, you can have a model where the universe size is less than or equal to 3.
- Existence of Models with Fewer Elements: Thus, since φ has a model with 7 elements, it implies that it can be satisfied with fewer elements, making it necessary that at least one model exists with a universe of size less than or equal to 3.
Conclusion
Therefore, the correct answer is option A: There exists at least one model of φ with a universe of size less than or equal to 3. This aligns with the flexibility allowed by the existential quantifiers and the nature of quantifier-free formulas.
The first-order logic sentence φ states that there exist elements s, t, u in the universe such that for all elements v, w, x, and y, the formula ψ(s, t, u, v, w, x, y) holds.
Key Insights
- Quantifier-Free Formula: Since ψ is quantifier-free, it can be satisfied in a certain configuration of elements in the universe.
- Existential and Universal Quantifiers: The existential quantifiers for s, t, and u indicate that we only need to find values for these three variables. The universal quantifiers for v, w, x, and y imply that the formula must hold for every possible combination of these remaining variables.
Reasons Why Option A is True
- Non-uniqueness of Models: The existence of a model with 7 elements does not restrict the number of elements in other possible models.
- Smaller Universes: It is possible to construct models of φ with fewer than 7 elements. For instance, if you can satisfy ψ with just 3 distinct elements, you can have a model where the universe size is less than or equal to 3.
- Existence of Models with Fewer Elements: Thus, since φ has a model with 7 elements, it implies that it can be satisfied with fewer elements, making it necessary that at least one model exists with a universe of size less than or equal to 3.
Conclusion
Therefore, the correct answer is option A: There exists at least one model of φ with a universe of size less than or equal to 3. This aligns with the flexibility allowed by the existential quantifiers and the nature of quantifier-free formulas.
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Consider the first-order logic sentenceφ ≡ ∃s∃t∃u∀v∀w∀x∀y ψ(s, t, u, v, w, x, y)where ψ(s, t, u, v, w, x, y) is a quantifier-free first-order logic formula using only predicate symbols, and possibly equality, but no function symbols. Suppose φ has a model with a universe containing 7 elements. Which one of the following statements is necessarily true?a)There exists at least one model of φ with universe of size less than or equal to 3b)There exists no model of φ with universe of size less than or equal to 3c)There exists no model of φ with universe size of greater than 7d)Every model of φ has a universe of size equal to 7Correct answer is option 'A'. Can you explain this answer?
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Consider the first-order logic sentenceφ ≡ ∃s∃t∃u∀v∀w∀x∀y ψ(s, t, u, v, w, x, y)where ψ(s, t, u, v, w, x, y) is a quantifier-free first-order logic formula using only predicate symbols, and possibly equality, but no function symbols. Suppose φ has a model with a universe containing 7 elements. Which one of the following statements is necessarily true?a)There exists at least one model of φ with universe of size less than or equal to 3b)There exists no model of φ with universe of size less than or equal to 3c)There exists no model of φ with universe size of greater than 7d)Every model of φ has a universe of size equal to 7Correct answer is option 'A'. Can you explain this answer? for Computer Science Engineering (CSE) 2024 is part of Computer Science Engineering (CSE) preparation. The Question and answers have been prepared according to the Computer Science Engineering (CSE) exam syllabus. Information about Consider the first-order logic sentenceφ ≡ ∃s∃t∃u∀v∀w∀x∀y ψ(s, t, u, v, w, x, y)where ψ(s, t, u, v, w, x, y) is a quantifier-free first-order logic formula using only predicate symbols, and possibly equality, but no function symbols. Suppose φ has a model with a universe containing 7 elements. Which one of the following statements is necessarily true?a)There exists at least one model of φ with universe of size less than or equal to 3b)There exists no model of φ with universe of size less than or equal to 3c)There exists no model of φ with universe size of greater than 7d)Every model of φ has a universe of size equal to 7Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for Computer Science Engineering (CSE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the first-order logic sentenceφ ≡ ∃s∃t∃u∀v∀w∀x∀y ψ(s, t, u, v, w, x, y)where ψ(s, t, u, v, w, x, y) is a quantifier-free first-order logic formula using only predicate symbols, and possibly equality, but no function symbols. Suppose φ has a model with a universe containing 7 elements. Which one of the following statements is necessarily true?a)There exists at least one model of φ with universe of size less than or equal to 3b)There exists no model of φ with universe of size less than or equal to 3c)There exists no model of φ with universe size of greater than 7d)Every model of φ has a universe of size equal to 7Correct answer is option 'A'. Can you explain this answer?.
Consider the first-order logic sentenceφ ≡ ∃s∃t∃u∀v∀w∀x∀y ψ(s, t, u, v, w, x, y)where ψ(s, t, u, v, w, x, y) is a quantifier-free first-order logic formula using only predicate symbols, and possibly equality, but no function symbols. Suppose φ has a model with a universe containing 7 elements. Which one of the following statements is necessarily true?a)There exists at least one model of φ with universe of size less than or equal to 3b)There exists no model of φ with universe of size less than or equal to 3c)There exists no model of φ with universe size of greater than 7d)Every model of φ has a universe of size equal to 7Correct answer is option 'A'. Can you explain this answer? for Computer Science Engineering (CSE) 2024 is part of Computer Science Engineering (CSE) preparation. The Question and answers have been prepared according to the Computer Science Engineering (CSE) exam syllabus. Information about Consider the first-order logic sentenceφ ≡ ∃s∃t∃u∀v∀w∀x∀y ψ(s, t, u, v, w, x, y)where ψ(s, t, u, v, w, x, y) is a quantifier-free first-order logic formula using only predicate symbols, and possibly equality, but no function symbols. Suppose φ has a model with a universe containing 7 elements. Which one of the following statements is necessarily true?a)There exists at least one model of φ with universe of size less than or equal to 3b)There exists no model of φ with universe of size less than or equal to 3c)There exists no model of φ with universe size of greater than 7d)Every model of φ has a universe of size equal to 7Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for Computer Science Engineering (CSE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the first-order logic sentenceφ ≡ ∃s∃t∃u∀v∀w∀x∀y ψ(s, t, u, v, w, x, y)where ψ(s, t, u, v, w, x, y) is a quantifier-free first-order logic formula using only predicate symbols, and possibly equality, but no function symbols. Suppose φ has a model with a universe containing 7 elements. Which one of the following statements is necessarily true?a)There exists at least one model of φ with universe of size less than or equal to 3b)There exists no model of φ with universe of size less than or equal to 3c)There exists no model of φ with universe size of greater than 7d)Every model of φ has a universe of size equal to 7Correct answer is option 'A'. Can you explain this answer?.
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