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{\bf\large {Worksheet 1: Sets and Logic}}
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\begin{enumerate} 
\item Which of the following sentences are \emph{statements} in the mathematical sense? For the ones that are statements, can you decide whether they are True or False? 
\begin{enumerate}[(a)]
\item It is sunny outside right now.
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\item Tomorrow the weather will be nice. 
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\item The $100$th digit of the decimal expansion of $\pi$ is $7$. 
%seems false-- looks like it's 9, after some googling. 
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\item The digits of $\pi$ encode the meaning of the Universe. 
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\item This statement is False. 
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\item This statement is True. 
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\item For any consistent system of axioms, there exists a statement about natural numbers that is true, but unprovable from these axioms.
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\item For some prime numbers $p$, the number $p+2$ is also prime. 
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\item For all prime numbers $p$, the number $p+2$ is also prime.
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\item There exist infinitely many primes $p$ such that the number $p+2$ is also prime. 


\end{enumerate} 

\newpage


\item  Are the following sets empty or not? When not empty, draw the set.  
\begin{enumerate}[(a)] 
\item  The set of all $x\in \R$ such that $x^2>4$ and $x<0$.
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\item  The set of all $x\in \R$ such that $x^2>4$ and $|x|<2$.
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\item  The set of all $x\in \R$ such that $x^2\ge 4$ and $|x|\le 2$.
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\item The set of all $(x,y)\in \R^2$ such that $x^2+y^2=1$ and $x<0$. 
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\item $\{(x,y)\in \R^2: \, x^2+y^2=-1\}$.
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\item $\{(x,y)\in \R^2: \, x^2-y^2=-1\}$.
\end{enumerate}
\end{enumerate}

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