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\author{Logan Axon}
\institute{Gonzaga University}
\date{\today}
\title{Logic basics}

\newtheorem{thm}{Theorem}
\theoremstyle{definition}
\newtheorem{dfn}{Definition}
\newtheorem{exercise}{Exercise}
\newtheorem{idea}{Idea}
\newtheorem{question}{Question}

\newcommand{\cc}{\mathcal{C}}
\newcommand{\rr}{\mathcal{R}}
\newcommand{\pp}{\mathbf{P}}
\newcommand{\N}{\text{N}}

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\begin{document}
%%%%%%
\begin{frame}{Statements and open sentences}

Statements: 
\begin{itemize}
\item 2 is an even integer.
\item 3 is an even integer.
\item 4 is an even integer.
\item \dots
\end{itemize}

\pause

\vspace{.6cm}

$P(x)$: $x$ is an even integer.

\pause

$P(x)$ is \textbf{not a statement}; it is an open sentence (with a free variable).

$P(7)$ is a statement.

\pause

\vspace{.6cm}

$R(f,g)$: $f$ is the derivative of $g$.

\end{frame}
%%%%%%
\begin{frame}{Famous statements}

Goldbach's conjecture (1742): every even integer greater than 2 is the sum of two prime numbers.

\vspace{.6cm}

Twin primes conjecture: there are infinitely many primes $p$ such that $p+2$ is also prime.

\vspace{.6cm}

Riemann hypothesis: the nontrivial zeros of the Riemann zeta function have real part equal to $\frac{1}{2}$.

\vspace{.6cm}

Banach-Tarski paradox (1924): A (solid) sphere may be decomposed into finitely many sets which can be rearranged to form two spheres, each of which is just as large as the original sphere. 
\url{https://youtu.be/s86-Z-CbaHA}

\end{frame}
%%%%%%
\begin{frame}{Examples}

\begin{enumerate}

\item If you eat your vegetables, then you'll get dessert.

\pause

\vspace{.6cm}

\item If you got dessert, then you must have eaten your vegetables.

\pause

\vspace{.6cm}

\item If you are 20 years old, then you were born in 1998 or 1997.

\pause

\vspace{.6cm}

\item If you were born in 1998 or 1997, then you are 20 years old.

\pause

\vspace{.6cm}

\item An integer is even if and only if it is divisible by 2.

\end{enumerate}

\end{frame}
%%%%%%
\begin{frame}{Vocabulary}

\begin{dfn} 
In the statment $P \implies Q$, $P$ is the \textbf{antecedent} and $Q$ is the \textbf{consequent}.
\end{dfn}

\begin{enumerate}
\item If the antecedent is true, then the consequent must also be true.

\item Converse?

\item If the antecedent is false, then the statement is true regardless of the consequent.
\end{enumerate}

\end{frame}
%%%%%%
\begin{frame}{Quantifiers}

Open sentence $P(x)$.

Statements:
\begin{itemize}
\item $\forall x, P(x)$ ``for all x, $P(x)$''
\item $\exists x, P(X)$ ``there is an x such that $P(x)$''
\end{itemize}

\end{frame}
%%%%%%
\begin{frame}{Examples involving quantifiers}

\begin{enumerate}

\item For every real number $x$, $x^2 > 0$.

\pause

\vspace{.6cm}

\item The polynomial $x^3 + x^2 + x +1$ has a real root.

\pause

\vspace{.6cm}

\item Every degree three polynomial has a real root.

\pause

\vspace{.6cm}

\item $\lim_{x \to a} f(x) = L$ if and only if for every number $\epsilon > 0$ there is a  number $\delta >0$ such that \[ \lvert x - a \rvert < \delta \implies \lvert f(x) - L \rvert < \epsilon. \]

\end{enumerate}

\end{frame}
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\end{document}