I would like some guidance on how to prove completeness/incompleteness for parts c-f. If I disprove, I'd like to use a counterexample to show that a formula is a tautology and yet not derivable from a set.If I prove, I'd like to show that for all possible formulas and sets, if the formula is a tautology to the set, then it is also derivable from the set. 2. Determine whether each of the following proposed proof systems are complete. Justify your answers.(a) Axioms: All formulas.Rules: None.(b) Axioms: None.Rules: From nothing, infer any formula(c) Axioms: All formulas of the form ( → $).Rules: Hypothetical Syllogism.(d) Axioms: All formulas of the form ( → 4).Rules: Modus Ponens.(e) Axioms: All formulas of the form (o → 4).Rules: Hypothetical Syllogism.(f) Axioms: All tautologies.Rules: From any o that is not a tautology, infer any formula.

Since, you have posted a multi parts questions but according to guidelines we will solve first three…

Answered: 2. Determine whether each of the…

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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I would like some guidance on how to prove completeness/incompleteness for parts c-f.

If I disprove, I'd like to use a counterexample to show that a formula is a tautology and yet not derivable from a set.

If I prove, I'd like to show that for all possible formulas and sets, if the formula is a tautology to the set, then it is also derivable from the set.

2. Determine whether each of the following proposed proof systems are complete. Justify your answers.
(a) Axioms: All formulas.
Rules: None.
(b) Axioms: None.
Rules: From nothing, infer any formula
(c) Axioms: All formulas of the form ( → $).
Rules: Hypothetical Syllogism.
(d) Axioms: All formulas of the form ( → 4).
Rules: Modus Ponens.
(e) Axioms: All formulas of the form (o → 4).
Rules: Hypothetical Syllogism.
(f) Axioms: All tautologies.
Rules: From any o that is not a tautology, infer any formula.

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