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\begin{document}
\begin{center}
{\bf \Large \bf CSC240 Winter 2021 Midterm Assessment Question 3}\\
YOUR NAME and STUDENT NUMBER
\end{center}
\medskip
\begin{enumerate}
\setcounter{enumi}{2}
\item
\begin{enumerate}
\item
\begin{question}
(2 marks)
Give a recursive definition of the set ${\mathcal S}_P$ of propositional formulas that can formed
using the ternary predicate $P:\{\True,\False\}^3 \rightarrow \{\True,\False\}$
and the propositional variable $X$.
\end{question}
\begin{solution}
{\bf Solution}:
\end{solution}
\item
\begin{question}
(8 marks)
Let $M:\{\True,\False\}^3 \rightarrow \{\True,\False\}$ be the ternary predicate that is true when 0 or 1 of its arguments are true
and is false when 2 or 3 of its arguments are true.\\
Use structural induction to prove that every formula in ${\mathcal S}_M$ is logically equivalent to $X$ or is logically equivalent to $\Not(X)$.
\end{question}
%The LaTeX symbol $\equiv$ may be useful.
\begin{solution}
{\bf Solution}:
\end{solution}
\item
\begin{question}
(8 marks)
Let $N:\{\True,\False\}^3 \rightarrow \{\True,\False\}$ be the ternary predicate that is true when 0 of its arguments are true
and is false when at least 1 of its arguments is true.\\
Prove that every unary predicate $U:\{\True,\False\} \rightarrow \{\True,\False\}$
is logically equivalent to some formula in ${\mathcal S}_N$.
\end{question}
\begin{solution}
{\bf Solution}:
\end{solution}
\end{enumerate}
\end{enumerate}
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