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In the following wff, there are six occurrences of x. From left to right, which most accurately represents if each of the six occurrences of $x$ are bound or free? Exp(x) ^ q(x) → 3x(p(x) ^ q(x)) b,b,b,b,b,f b,b,b,b,b,b b,b,f,b,b,b

In the following wff, there are six occurrences of x. From left to right, which most accurately represents if each of the six occurrences of $x$ are bound or free? Exp(x) ^ q(x) → 3x(p(x) ^ q(x)) b,b,b,b,b,f b,b,b,b,b,b b,b,f,b,b,b

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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In the following well-formed formula (wff), there are six occurrences of \( x \). From left to right, which most accurately represents if each of the six occurrences of \( x \) are bound or free? \[ \exists x p(x) \land q(x) \rightarrow \exists x (p(x) \land q(x)) \] Options: - ○ b, b, b, b, b, f - ○ b, b, b, b, b, b - ○ b, f, b, b, b, b - ○ b, b, f, b, b, f Explanation: - "b" stands for "bound" and "f" stands for "free". - The proposition involves logical quantifiers and predicates with \( x \) as a variable whose occurrences are needed to be identified as bound or free. - The aim is to determine the status of each occurrence of \( x \) in the formula based on the scope of quantifiers.
Transcribed Image Text:In the following well-formed formula (wff), there are six occurrences of \( x \). From left to right, which most accurately represents if each of the six occurrences of \( x \) are bound or free? \[ \exists x p(x) \land q(x) \rightarrow \exists x (p(x) \land q(x)) \] Options: - ○ b, b, b, b, b, f - ○ b, b, b, b, b, b - ○ b, f, b, b, b, b - ○ b, b, f, b, b, f Explanation: - "b" stands for "bound" and "f" stands for "free". - The proposition involves logical quantifiers and predicates with \( x \) as a variable whose occurrences are needed to be identified as bound or free. - The aim is to determine the status of each occurrence of \( x \) in the formula based on the scope of quantifiers.
**Logical Reasoning Quiz: Validity and Satisfiability of WFFs** ### Which one of these wffs (well-formed formulas) is **valid**? - **Option A:** \( p(c) \land \exists x \lnot p(x) \) - **Option B:** \( \forall x p(x) \to \exists x p(x) \) - **Option C:** \( \exists x (p(x) \land \lnot p(x)) \) - **Option D:** \( \exists x p(x) \to \forall x p(x) \) ### Which one of these wffs is **unsatisfiable**? - **Option A:** \( \exists x p(x) \to \forall x p(x) \) - **Option B:** \( p(c) \land \exists x \lnot p(x) \) - **Option C:** \( \forall x p(x) \to \exists x p(x) \) - **Option D:** \( \exists x (p(x) \land \lnot p(x)) \) ### Explanation: - **Validity:** A wff is considered valid if it is true in every possible interpretation or model. - **Unsatisfiability:** A wff is unsatisfiable if there is no interpretation or model in which the formula can be true. Use this quiz to test your understanding of logical validity and satisfiability in predicate logic. Analyze each option and determine which one fits the criteria of being valid or unsatisfiable.
Transcribed Image Text:**Logical Reasoning Quiz: Validity and Satisfiability of WFFs** ### Which one of these wffs (well-formed formulas) is **valid**? - **Option A:** \( p(c) \land \exists x \lnot p(x) \) - **Option B:** \( \forall x p(x) \to \exists x p(x) \) - **Option C:** \( \exists x (p(x) \land \lnot p(x)) \) - **Option D:** \( \exists x p(x) \to \forall x p(x) \) ### Which one of these wffs is **unsatisfiable**? - **Option A:** \( \exists x p(x) \to \forall x p(x) \) - **Option B:** \( p(c) \land \exists x \lnot p(x) \) - **Option C:** \( \forall x p(x) \to \exists x p(x) \) - **Option D:** \( \exists x (p(x) \land \lnot p(x)) \) ### Explanation: - **Validity:** A wff is considered valid if it is true in every possible interpretation or model. - **Unsatisfiability:** A wff is unsatisfiable if there is no interpretation or model in which the formula can be true. Use this quiz to test your understanding of logical validity and satisfiability in predicate logic. Analyze each option and determine which one fits the criteria of being valid or unsatisfiable.
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