\part{Fundamentals}


\chapter{Propositional logic}


This part of the book provides an understanding of logic that is used as the language for mathematics as well as demonstrate how logic is used in the art of proofs.

When sufficient mathematics is developed (formal language theory, set theory, abstract algebra, etc.) one can then study the properties of logic in a deeper sense. The objective of this part of the book is to understand the fundamentals of logic, its algebraic behaviour, and how it is used in mathematics.




\section{Terms}
\subsection{Variables}

\begin{definition}[Propositional variable]
A \emph{propositional variable} is a symbol representing some free variable that takes a single truth value (either the top or the bottom). It is a type of \emph{atomic formula}; a propositional formula irreducible to constituent parts (more on this later).
They are often represented by minuscule letters $p,q,\hdot$
\end{definition}

%\subsection{Function}
\section{Logical connectives and truth tables}


\subsection{Nullary connectives (Truth values)}
Value that relates a proposition to its truth.
\begin{definition}[Top]
The \emph{top} or \emph{verum} is the symbol $\top$ repesenting truth.
\end{definition}

\begin{definition}[Bottom]
The \emph{bottom} or \emph{falsum} is the symbol $\bot$ repesenting nontruth.
\end{definition}

\subsection{Unary connectives}
\begin{definition}[Negation (NOT)]
\[\neg P \]
\end{definition}

\subsection{Binary connectives}
\begin{definition}[Conjunction (AND)]
\[P \land Q\]
\end{definition}

\begin{definition}[Disjunction (OR)]
\[P \lor Q\]
\end{definition}

\begin{definition}[Conditional (IMPLY)]
\[P \implies Q \]
\end{definition}

\begin{definition}[Biconditional (IFF)]
\[P \iff Q \]
\end{definition}



\section{Tautologies and contradictions}


\begin{definition}[Formula]
\end{definition}

\begin{definition}[Tautology]
A tautology is a formula that is true under any propositonal variable setting
\end{definition}

\begin{definition}[Contradiction]
A contradiction is a formula that is false under any propositonal variable setting
\end{definition}









\chapter{First-order logic}

In propositional logic, the most basic constituents were propositional variables, however what if we consider predicates? Predicates are functions that map objects to truth variables. Such a logic is called \emph{first order logic}, and we'll introduce some more logical symbols to discuss what statements are possible with predicates.




\section{Equality}

\section{Quantifiers}


\subsection{Universal quantifier}
\begin{definition}[Universal quantifier (FORALL)]
\[\forall x \]
\end{definition}

\subsection{Existential quantifier}
\begin{definition}[Existential quantifier (EXISTS)]
\[\exists x \]
\end{definition}

\[\exists! x \]

\subsection{De Morgan's laws for quantifiers}







\chapter{Proof techniques}

It is a useful skill for a mathematician to leverage knowledge of logic to create proofs. 

Indeed, the key to creating proofs for various theorems is to find a suitable proof method.

\section{Direct proof}
\section{Proof by contraposition}
\section{Proof by contradiction}
\section{Proof by mathematical induction}
\section{Combinatorial proof}
\section{Proof by construction}