GATE CS 2018 Set 1 — Question 38
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Engineering Mathematics → Mathematical Logic → Predicate Logic & Quantifiers
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Question
Consider the first-order logic sentencewhere is a quantifier-free first-order logic formula using only predicate symbols, and possibly equality, but no function symbols. Suppose has a model with a universe containing 7 elements.Which one of the following statements is necessarily true?
Correct answer
(A) There exists at least one model of φ with universe of size less than or equal to 3.
Solution
The sentence is of the form , where is quantifier-free and contains no function symbols. If has a model with a universe of size 7, then there exist elements such that for all , the formula is true in .Consider a sub-universe . The size of is at most 3 (it could be less if are not distinct). Since is quantifier-free and there are no function symbols, the truth of for elements in depends only on the relations defined on those elements. Because holds for all elements in the larger universe , it must also hold for all elements in the subset . Thus, the structure restricted to is also a model of . Therefore, there exists a model of size . Option (A) is necessarily true.
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