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\begin{center}
\textbf{\Large
              AIT Semantic and Declarative Technologies Course\\[1ex]
              Practice L2:  Propositional Logic -- expressing Boolean functions \\[1ex]
              }
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\begin{enumerate}[itemsep=2pt,parsep=0pt,leftmargin=*]

\item Implication (Recall that $U \rightarrow V = \neg U \vee V$.)

\begin{enumerate}
\item Express implication using negation and conjunction only:

$
 (U \rightarrow V) = $

\vspace*{5ex}

\item Express disjunction using implication and negation only:

$
 (U \vee V) = $

\vspace*{5ex}



\end{enumerate}



\item Equivalence


\begin{enumerate}
\item Express equivalence using implication and conjunction:

$
 (U \leftrightarrow V) = $

\vspace*{5ex}

\item Express equivalence using negation, disjunction and conjunction:

$
 (U \leftrightarrow V) = $

\vspace*{5ex}


\end{enumerate}


\item Exclusive Or

\begin{enumerate}
\item Express ``exclusive or'' ($\oplus$) using negation and equivalence only:

$
 (U \oplus V) = $

\vspace*{5ex}

\item Express ``exclusive or'' using negation, conjunction and disjunction:

$
 (U \oplus V) = $

\vspace*{5ex}

\end{enumerate}

\item  Complex Boolean expressions (you can use arbitrary operations
  introduced above, try to find the simplest formula)
    
  \begin{enumerate}
    \item Express the condition that exactly one of the propositions $A$
      and $B$ are true

      \vspace*{7ex}
      
    \item Express the condition that exactly two of the propositions $A$
      and $B$ and $C$ are true
      
      \vspace*{7ex}
      
    \item Express the boolean if-then-else function, similar to the C
      language  expression \texttt{(x ? y : z)}:

\begin{equation*}
  (\text{if } A \text{ then } B \text{ else } C) =
  \begin{cases}
    B & \text{if $A$ is true} \\
    C & \text{otherwise}
  \end{cases}
\end{equation*}

\vspace*{7ex}      

\end{enumerate}
\end{enumerate}



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