For each of the following sequents, either provide a natural deduction proof, or a counter-example that proves the sequent invalid. \\

\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. \\

\noindent For counterexamples, give a complete model. Show that the model satisfies
the premise(s) of the sequent in question, but does not satisfy the respective conclusion.
For each of the following sequents, either provide a natural deduction proof, or a counter-example that proves the sequent invalid.

\begin{questionSection}{Rules for Natural Deduction}
\question{natural_deduction_propositional_logic/0038}
		     {natural_deduction_propositional_logic/0038_sol}
		     {3cm}

\mcquestion{natural_deduction_propositional_logic/0053}
		       {natural_deduction_propositional_logic/0053_sol}

\question{natural_deduction_propositional_logic/0041}
         {natural_deduction_propositional_logic/0041_sol}
         {3cm}
\question{natural_deduction_propositional_logic/0043}
         {natural_deduction_propositional_logic/0043_sol}
         {3cm}
\question{natural_deduction_propositional_logic/0042}
         {natural_deduction_propositional_logic/0042_sol}
         {3cm}
%\question{natural_deduction_propositional_logic/0049}
%         {no_solution}
%         {3cm}
%\question{natural_deduction_propositional_logic/0044}
%         {natural_deduction_propositional_logic/0044_sol}
%         {3cm}

%\question{natural_deduction_propositional_logic/0047}
%         {no_solution}
%         {3cm}
%\question{natural_deduction_propositional_logic/0048}
%         {no_solution}
%         {3cm}
\question{natural_deduction_propositional_logic/0004}
         {natural_deduction_propositional_logic/0004_sol}
         {6cm}

\question{natural_deduction_propositional_logic/0079}
         {natural_deduction_propositional_logic/0079_sol}
         {6cm}

\question{natural_deduction_propositional_logic/0006}
         {natural_deduction_propositional_logic/0006_sol}
         {3cm}


\question{natural_deduction_propositional_logic/0080}
         {natural_deduction_propositional_logic/0080_sol}
         {6cm}

\question{natural_deduction_propositional_logic/0081}
         {natural_deduction_propositional_logic/0081_sol}
         {6cm}

\question{natural_deduction_propositional_logic/0008}
         {natural_deduction_propositional_logic/0008_sol}
         {6cm}

\question{natural_deduction_propositional_logic/0009}
         {natural_deduction_propositional_logic/0009_sol}
         {3cm}

\question{natural_deduction_propositional_logic/0082}
         {natural_deduction_propositional_logic/0082_sol}
         {3cm}

\question{natural_deduction_propositional_logic/0083}
         {natural_deduction_propositional_logic/0083_sol}
         {3cm}

\question{natural_deduction_propositional_logic/0084}
         {natural_deduction_propositional_logic/0084_sol}
         {3cm}

\question{natural_deduction_propositional_logic/0012}
         {natural_deduction_propositional_logic/0012_sol}
         {6cm}

\question{natural_deduction_propositional_logic/0011}
         {natural_deduction_propositional_logic/0011_sol}
         {6cm}

\question{natural_deduction_propositional_logic/0010}
         {natural_deduction_propositional_logic/0010_sol}
         {6cm}

\question{natural_deduction_propositional_logic/0086}
         {natural_deduction_propositional_logic/0086_sol}
         {6cm}

\question{natural_deduction_propositional_logic/1007}
		     {natural_deduction_propositional_logic/1007_sol}
		     {3cm}
\question{natural_deduction_propositional_logic/1010}
		     {natural_deduction_propositional_logic/1010_sol}
		     {3cm}


%\question{natural_deduction_propositional_logic/1003}
%		     {natural_deduction_propositional_logic/1003_sol}
%		     {3cm}
\question{natural_deduction_propositional_logic/1004}
		     {natural_deduction_propositional_logic/1004_sol}
		     {3cm}
%\question{natural_deduction_propositional_logic/1006}
%		     {natural_deduction_propositional_logic/1006_sol}
%		     {3cm}
%\question{natural_deduction_propositional_logic/1005}
%		     {natural_deduction_propositional_logic/1005_sol}
%		     {3cm}
\question{natural_deduction_propositional_logic/1009}
		     {natural_deduction_propositional_logic/1009_sol}
		     {3cm}
\question{natural_deduction_propositional_logic/0040}
         {natural_deduction_propositional_logic/0040_sol}
         {3cm}

%\question{natural_deduction_propositional_logic/0016}
%         {natural_deduction_propositional_logic/0016_sol}
%         {3cm}

\question{natural_deduction_propositional_logic/0017}
         {natural_deduction_propositional_logic/0017_sol}
         {3cm}


%\question{natural_deduction_propositional_logic/0018}
%         {no_solution}
%         {3cm}

\question{natural_deduction_propositional_logic/0019}
         {natural_deduction_propositional_logic/0019_sol}
         {3cm}


\question{natural_deduction_propositional_logic/0020}
         {natural_deduction_propositional_logic/0020_sol}
         {3cm}

\question{natural_deduction_propositional_logic/0021}
         {natural_deduction_propositional_logic/0021_sol}
         {3cm}

\question{natural_deduction_propositional_logic/0022}
         {natural_deduction_propositional_logic/0022_sol}
         {3cm}

\question{natural_deduction_propositional_logic/0023}
         {natural_deduction_propositional_logic/0023_sol}
         {3cm}

\question{natural_deduction_propositional_logic/0024}
         {natural_deduction_propositional_logic/0024_sol}
         {3cm}

\question{natural_deduction_propositional_logic/0025}
         {natural_deduction_propositional_logic/0025_sol}
         {3cm}



\question{natural_deduction_propositional_logic/0027}
         {natural_deduction_propositional_logic/0027_sol}
         {3cm}




\question{natural_deduction_propositional_logic/0028}
         {natural_deduction_propositional_logic/0028_sol}
         {3cm}

\question{natural_deduction_propositional_logic/0029}
         {natural_deduction_propositional_logic/0029_sol}
         {3cm}


\question{natural_deduction_propositional_logic/0030}
         {natural_deduction_propositional_logic/0030_sol}
         {3cm}

\question{natural_deduction_propositional_logic/0031}
         {natural_deduction_propositional_logic/0031_sol}
         {3cm}

\question{natural_deduction_propositional_logic/0032}
         {natural_deduction_propositional_logic/0032_sol}
         {3cm}



\question{natural_deduction_propositional_logic/0007}
         {natural_deduction_propositional_logic/0007_sol}
         {3cm}

\question{natural_deduction_propositional_logic/0059}
         {natural_deduction_propositional_logic/0059_sol}
         {3cm}

\question{natural_deduction_propositional_logic/0060}
         {natural_deduction_propositional_logic/0060_sol}
         {3cm}

\question{natural_deduction_propositional_logic/0061}
         {natural_deduction_propositional_logic/0061_sol}
         {3cm}

\question{natural_deduction_propositional_logic/0062}
         {natural_deduction_propositional_logic/0062_sol}
         {3cm}

\question{natural_deduction_propositional_logic/0063}
         {natural_deduction_propositional_logic/0063_sol}
         {3cm}

\question{natural_deduction_propositional_logic/0064}
         {natural_deduction_propositional_logic/0064_sol}
         {3cm}

\question{natural_deduction_propositional_logic/0065}
         {natural_deduction_propositional_logic/0065_sol}
         {3cm}

\question{natural_deduction_propositional_logic/0066}
         {natural_deduction_propositional_logic/0066_sol}
         {3cm}

\question{natural_deduction_propositional_logic/0067}
         {natural_deduction_propositional_logic/0067_sol}
         {3cm}

\question{natural_deduction_propositional_logic/0068}
         {natural_deduction_propositional_logic/0068_sol}
         {3cm}

\question{natural_deduction_propositional_logic/0069}
         {natural_deduction_propositional_logic/0069_sol}
         {3cm}

\question{natural_deduction_propositional_logic/0070}
         {natural_deduction_propositional_logic/0070_sol}
         {3cm}

\question{natural_deduction_propositional_logic/0071}
         {natural_deduction_propositional_logic/0071_sol}
         {3cm}

\question{natural_deduction_propositional_logic/0072}
         {natural_deduction_propositional_logic/0072_sol}
         {3cm}

\question{natural_deduction_propositional_logic/0073}
         {natural_deduction_propositional_logic/0073_sol}
         {3cm}

\question{natural_deduction_propositional_logic/0074}
         {natural_deduction_propositional_logic/0074_sol}
         {3cm}

\question{natural_deduction_propositional_logic/0076}
         {natural_deduction_propositional_logic/0076_sol}
         {3cm}

\question{natural_deduction_propositional_logic/0078}
         {natural_deduction_propositional_logic/0078_sol}
         {3cm}


\question{natural_deduction_propositional_logic/0033}
         {natural_deduction_propositional_logic/0033_sol}
         {3cm}

\question{natural_deduction_propositional_logic/0035}
         {natural_deduction_propositional_logic/0035_sol}
         {3cm}

\question{natural_deduction_propositional_logic/0036}
         {natural_deduction_propositional_logic/0036_sol}
         {3cm}

\question{natural_deduction_propositional_logic/0037}
         {natural_deduction_propositional_logic/0037_sol}
         {3cm}
\question{natural_deduction_propositional_logic/1001}
		     {natural_deduction_propositional_logic/1001_sol}
		     {3cm}
\question{natural_deduction_propositional_logic/1002}
		     {natural_deduction_propositional_logic/1002_sol}
		     {3cm}
\question{natural_deduction_propositional_logic/0005}
         {natural_deduction_propositional_logic/0005_sol}
         {3cm}
\question{natural_deduction_propositional_logic/0075}
         {natural_deduction_propositional_logic/0075_sol}
         {3cm}
\question{natural_deduction_propositional_logic/0077}
         {natural_deduction_propositional_logic/0077_sol}
         {3cm}

%\question{natural_deduction_propositional_logic/0013}
%         {natural_deduction_propositional_logic/0013_sol}
%         {6cm}
%         \ifsolution
%           \leavevmode \newline \vspace{1mm} \noindent \\
%           \widefbox{\input{natural_deduction_propositional_logic/0013_sol_alternative.tex}}
%         \fi
\question{natural_deduction_propositional_logic/0014}
         {natural_deduction_propositional_logic/0014_sol}
         {3cm}
\question{natural_deduction_propositional_logic/0015}
         {natural_deduction_propositional_logic/0015_sol}
         {3cm}
\end{questionSection}

\begin{questionSection}{Soundness and Completeness of Natural Deduction}

\question{natural_deduction_propositional_logic/0045}
         {natural_deduction_propositional_logic/0045_sol}
         {3cm}

\question{natural_deduction_propositional_logic/0046}
         {natural_deduction_propositional_logic/0046_sol}
         {3cm}


\question{natural_deduction_propositional_logic/0050}
         {no_solution}
         {3cm}

\mcquestion{natural_deduction_propositional_logic/0056}
           {natural_deduction_propositional_logic/0056}

%\mcquestion{natural_deduction_propositional_logic/0057}
%           {natural_deduction_propositional_logic/0057}

%\mcquestion{natural_deduction_propositional_logic/0058}
%           {natural_deduction_propositional_logic/0058}

\question{natural_deduction_propositional_logic/0052}
         {no_solution}
         {3cm}


\end{questionSection}