Propositional calculus

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Problem: For the given propositional logic formulas, build the corresponding logic functions in the form of truth tables, determine the validity, satisfiability (non-satisfiability) and the number of models of the formula: \[ f(p, q)=p \&(q \vee \neg p) \&((\neg q \rightarrow p) \rightarrow q) . \]

6.1.1.1 Propositional calculus

1.53 $

Problem: Write the following statements in the form of propositional logic formulas, build a truth table and determine the validity, satisfiability (nonsatisfiability) and the number of models of the obtained formulas: "If the workers or the administration persist, the strike will be settled when and only when the government gets an injunction, but no troops are sent to the factory."

6.1.1.2 Propositional calculus

2.04 $

Problem: Prove the non-satisfiability (or satisfiability) of the following sets of clauses by using the resolution method. Apply an arbitrary order of enumeration of clauses, as well as, at the instruction of the teacher, one of the following strategies: preference for monomials, linear, level saturation. \[ \{(q \vee \neg \tau),(\neg q \vee \neg \tau),(q \vee \tau),(\neg p \vee \neg \tau), \neg q\} . \]

6.1.1.3 Propositional calculus

1.53 $

Problem: Write down formally the following reasoning in the language of propositional logic and prove its validity using the method of resolutions. Premise: wages will increase only if there is inflation. If there is inflation, the cost of living will increase. Conclusion: The cost of living will increase.

6.1.1.4 Propositional calculus

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Problem: The following notation is an expression (formula) of the propositional algebra: 1) \( ((A \wedge B) \Rightarrow C \) 2) \( A \Rightarrow \wedge B \Leftrightarrow C \); 3) \( A-B \wedge \bar{A} \); 4) \( A \vee(B \Rightarrow \bar{A}) \).

6.1.1.5 Propositional calculus

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Problem: Prove in propositional calculus (letters denote arbitrary formulas): \[ (A \rightarrow B) \rightarrow((C \vee(A \rightarrow C)) \vee B) . \]

6.1.1.6 Propositional calculus

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Problem: Prove in propositional calculus: \[ (F \supset G) \supset((F \supset \neg G) \supset \neg F) . \]

6.1.1.9 Propositional calculus

2.55 $