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349 lines
11 KiB
349 lines
11 KiB
For each of the following sequents, either provide a natural deduction proof, or a counter-example that proves the sequent invalid. \\
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\noindent For proofs, clearly indicate which rule, and what assumptions/premises/\\intermediate results you are using in each step. Also clearly indicate the scope of any boxes you use. \\
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\noindent For counterexamples, give a complete model. Show that the model satisfies
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the premise(s) of the sequent in question, but does not satisfy the respective conclusion.
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For each of the following sequents, either provide a natural deduction proof, or a counter-example that proves the sequent invalid.
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\begin{questionSection}{Rules for Natural Deduction}
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\question{natural_deduction_propositional_logic/0038}
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{natural_deduction_propositional_logic/0038_sol}
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{3cm}
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\mcquestion{natural_deduction_propositional_logic/0053}
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{natural_deduction_propositional_logic/0053_sol}
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\question{natural_deduction_propositional_logic/0041}
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{natural_deduction_propositional_logic/0041_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/0043}
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{natural_deduction_propositional_logic/0043_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/0042}
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{natural_deduction_propositional_logic/0042_sol}
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{3cm}
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%\question{natural_deduction_propositional_logic/0049}
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% {no_solution}
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% {3cm}
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%\question{natural_deduction_propositional_logic/0044}
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% {natural_deduction_propositional_logic/0044_sol}
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% {3cm}
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%\question{natural_deduction_propositional_logic/0047}
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% {no_solution}
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% {3cm}
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%\question{natural_deduction_propositional_logic/0048}
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% {no_solution}
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% {3cm}
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\question{natural_deduction_propositional_logic/0004}
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{natural_deduction_propositional_logic/0004_sol}
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{6cm}
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\question{natural_deduction_propositional_logic/0079}
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{natural_deduction_propositional_logic/0079_sol}
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{6cm}
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\question{natural_deduction_propositional_logic/0006}
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{natural_deduction_propositional_logic/0006_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/0080}
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{natural_deduction_propositional_logic/0080_sol}
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{6cm}
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\question{natural_deduction_propositional_logic/0081}
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{natural_deduction_propositional_logic/0081_sol}
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{6cm}
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\question{natural_deduction_propositional_logic/0008}
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{natural_deduction_propositional_logic/0008_sol}
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{6cm}
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\question{natural_deduction_propositional_logic/0009}
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{natural_deduction_propositional_logic/0009_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/0082}
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{natural_deduction_propositional_logic/0082_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/0083}
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{natural_deduction_propositional_logic/0083_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/0084}
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{natural_deduction_propositional_logic/0084_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/0012}
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{natural_deduction_propositional_logic/0012_sol}
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{6cm}
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\question{natural_deduction_propositional_logic/0011}
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{natural_deduction_propositional_logic/0011_sol}
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{6cm}
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\question{natural_deduction_propositional_logic/0010}
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{natural_deduction_propositional_logic/0010_sol}
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{6cm}
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\question{natural_deduction_propositional_logic/0086}
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{natural_deduction_propositional_logic/0086_sol}
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{6cm}
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\question{natural_deduction_propositional_logic/1007}
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{natural_deduction_propositional_logic/1007_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/1010}
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{natural_deduction_propositional_logic/1010_sol}
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{3cm}
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%\question{natural_deduction_propositional_logic/1003}
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% {natural_deduction_propositional_logic/1003_sol}
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% {3cm}
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\question{natural_deduction_propositional_logic/1004}
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{natural_deduction_propositional_logic/1004_sol}
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{3cm}
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%\question{natural_deduction_propositional_logic/1006}
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% {natural_deduction_propositional_logic/1006_sol}
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% {3cm}
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%\question{natural_deduction_propositional_logic/1005}
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% {natural_deduction_propositional_logic/1005_sol}
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% {3cm}
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\question{natural_deduction_propositional_logic/1009}
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{natural_deduction_propositional_logic/1009_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/0040}
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{natural_deduction_propositional_logic/0040_sol}
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{3cm}
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%\question{natural_deduction_propositional_logic/0016}
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% {natural_deduction_propositional_logic/0016_sol}
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% {3cm}
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\question{natural_deduction_propositional_logic/0017}
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{natural_deduction_propositional_logic/0017_sol}
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{3cm}
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%\question{natural_deduction_propositional_logic/0018}
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% {no_solution}
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% {3cm}
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\question{natural_deduction_propositional_logic/0019}
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{natural_deduction_propositional_logic/0019_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/0020}
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{natural_deduction_propositional_logic/0020_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/0021}
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{natural_deduction_propositional_logic/0021_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/0022}
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{natural_deduction_propositional_logic/0022_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/0023}
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{natural_deduction_propositional_logic/0023_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/0024}
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{natural_deduction_propositional_logic/0024_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/0025}
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{natural_deduction_propositional_logic/0025_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/0027}
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{natural_deduction_propositional_logic/0027_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/0028}
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{natural_deduction_propositional_logic/0028_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/0029}
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{natural_deduction_propositional_logic/0029_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/0030}
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{natural_deduction_propositional_logic/0030_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/0031}
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{natural_deduction_propositional_logic/0031_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/0032}
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{natural_deduction_propositional_logic/0032_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/0007}
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{natural_deduction_propositional_logic/0007_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/0059}
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{natural_deduction_propositional_logic/0059_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/0060}
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{natural_deduction_propositional_logic/0060_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/0061}
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{natural_deduction_propositional_logic/0061_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/0062}
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{natural_deduction_propositional_logic/0062_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/0063}
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{natural_deduction_propositional_logic/0063_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/0064}
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{natural_deduction_propositional_logic/0064_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/0065}
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{natural_deduction_propositional_logic/0065_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/0066}
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{natural_deduction_propositional_logic/0066_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/0067}
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{natural_deduction_propositional_logic/0067_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/0068}
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{natural_deduction_propositional_logic/0068_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/0069}
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{natural_deduction_propositional_logic/0069_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/0070}
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{natural_deduction_propositional_logic/0070_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/0071}
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{natural_deduction_propositional_logic/0071_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/0072}
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{natural_deduction_propositional_logic/0072_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/0073}
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{natural_deduction_propositional_logic/0073_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/0074}
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{natural_deduction_propositional_logic/0074_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/0076}
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{natural_deduction_propositional_logic/0076_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/0078}
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{natural_deduction_propositional_logic/0078_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/0033}
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{natural_deduction_propositional_logic/0033_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/0035}
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{natural_deduction_propositional_logic/0035_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/0036}
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{natural_deduction_propositional_logic/0036_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/0037}
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{natural_deduction_propositional_logic/0037_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/1001}
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{natural_deduction_propositional_logic/1001_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/1002}
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{natural_deduction_propositional_logic/1002_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/0005}
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{natural_deduction_propositional_logic/0005_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/0075}
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{natural_deduction_propositional_logic/0075_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/0077}
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{natural_deduction_propositional_logic/0077_sol}
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{3cm}
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%\question{natural_deduction_propositional_logic/0013}
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% {natural_deduction_propositional_logic/0013_sol}
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% {6cm}
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% \ifsolution
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% \leavevmode \newline \vspace{1mm} \noindent \\
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% \widefbox{\input{natural_deduction_propositional_logic/0013_sol_alternative.tex}}
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% \fi
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\question{natural_deduction_propositional_logic/0014}
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{natural_deduction_propositional_logic/0014_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/0015}
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{natural_deduction_propositional_logic/0015_sol}
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{3cm}
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\end{questionSection}
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\begin{questionSection}{Soundness and Completeness of Natural Deduction}
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\question{natural_deduction_propositional_logic/0045}
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{natural_deduction_propositional_logic/0045_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/0046}
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{natural_deduction_propositional_logic/0046_sol}
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{3cm}
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\question{natural_deduction_propositional_logic/0050}
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{no_solution}
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{3cm}
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\mcquestion{natural_deduction_propositional_logic/0056}
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{natural_deduction_propositional_logic/0056}
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%\mcquestion{natural_deduction_propositional_logic/0057}
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% {natural_deduction_propositional_logic/0057}
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%\mcquestion{natural_deduction_propositional_logic/0058}
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% {natural_deduction_propositional_logic/0058}
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\question{natural_deduction_propositional_logic/0052}
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{no_solution}
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{3cm}
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\end{questionSection}
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