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{\bf CSci 243 Homework 2}
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**My name**  %update me!!
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\begin{enumerate}

\item (3 points each) Let $P(x)$, $Q(x)$, and $R(x)$ be the statements ``$x$ is a professor'', ``$x$ is ignorant'', and ``$x$ is vain'', respectively. Express each of these statements using quantifiers; logical operations; and $P(x)$, $Q(x)$, and $R(x)$, where the domain consists all people.
\begin{enumerate}
    \item No professor are ignorant.
    \item All ignorant people are vain.
    \item No professors are vain.
\end{enumerate}

\item (2 points each) What are the truth values of these statements?
\begin{enumerate}
    \item $\exists!xP(x) \rightarrow \exists xP(x)$
    \item $\forall xP(x) \rightarrow \exists!xP(x)$
    \item $\exists!xP(x) \rightarrow \lnot \forall xP(x)$
\end{enumerate}

\item Understanding quantified predicates.
\begin{enumerate}
\item (3 points) English to quantified predicates: 
Use predicates, quantifiers, logical and mathematical operators to express statement ``There is a positive integer that is not the sum of three squares".
\item (7 points) Quantified predicate to English: 
Give the truth value of each of these statement if the domain of all variables consists of all real numbers. 
\begin{enumerate}
\item $\forall x\exists y (x=y^2)$
\item $\exists x\forall y (xy=0)$
\item $\forall x(x\not=0\rightarrow\exists y(xy=1))$
\item $\exists x\forall y(y\not=0\rightarrow xy=1)$
\item $\forall x\exists y (x+y=1)$
\item $\forall x\exists y(x+y=2\land 2x-y=1)$
\item $\forall x\forall y\exists z(z=(x+y)/2)$
\end{enumerate}
\end{enumerate}



\item (2 points each) Rewrite each of these statements so that negations appear only within predicates, i.e., so that no negation is outside a quantifier or an expression involving logical operators.
\begin{enumerate}
\item $\lnot\exists y\exists x P(x,y)$
\item $\lnot\forall x \exists y P(x,y)$
\item $\lnot\exists y(Q(y)\land \forall x\lnot R(x,y))$
\item $\lnot\exists y(\exists xR(x,y)\lor\forall x S(x,y))$
\item $\lnot\exists y(\forall x\exists z T(x,y,z)\lor \exists x\forall z U(x,y,z))$
\end{enumerate}


\end{enumerate}

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