logic_teml.tex

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\title{Homework 1}
\author{Juan Alvarado}
\maketitle

\begin{enumerate}

\item \textbf{Problem 1}: 
\textit{Order the functions by asymptotic notation.}\\

Use a truth table to prove the following logical equivalence.
\begin{displaymath}
  P \wedge (Q \vee R) \equiv (P \wedge Q) \vee (P \wedge R)
\end{displaymath}
Solution:

\begin{displaymath}  % start unumbered math environment
  %
  % Start a table in math mode.  The |c|c|c|c|c|c|c|c| string is a
  % format string that says there will be 8 colunms in the table.  The
  % c's indicate that the data in each column will be centered (use l
  % for left justified and r for right justified).  The vertical bar
  % means that lines will be drawn between columns.  The trailing
  % \hline causes a horizontal line to be drawn across the top of the
  % table.
  %
  \begin{array}{|c|c|c|c|c|c|c|c|}\hline
    %
    % Each row of the table consists of data separated by "&" symbols.
    % Each row must end with "\\" to cause a newline.  A trailing
    % \hline will cause a line to be drawn under the row.  A double
    % \hline is often used to separate the table header from the rest
    % of the table. 
    %
    P & Q & R & Q \vee R & P \wedge (Q \vee R) & P \wedge Q & P \wedge
    R & (P \wedge Q) \vee (P \wedge R)\\\hline\hline
    T & T & T & T & \mathbf{T} & T & T & \mathbf{T}\\\hline
    T & T & F & T & \mathbf{T} & T & F & \mathbf{T}\\\hline
    T & F & T & T & \mathbf{T} & F & T & \mathbf{T}\\\hline
    T & F & F & F & \mathbf{F} & F & F & \mathbf{F}\\\hline
    F & T & T & T & \mathbf{F} & F & F & \mathbf{F}\\\hline
    F & T & F & T & \mathbf{F} & F & F & \mathbf{F}\\\hline
    F & F & T & T & \mathbf{F} & F & F & \mathbf{F}\\\hline
    F & F & F & F & \mathbf{F} & F & F & \mathbf{F}\\\hline
  \end{array}
\end{displaymath}


\end{enumerate}




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