MathematicalConnectiveSymbols 2026-02-21 17:30:44 2026-02-21 19:07:50 Definition negation conjunction disjunction implication antecedent consequent biconditional truth tables double negation % this is the default PhysicsLibrary preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them \usepackage{geometry} \usepackage{hyperref} % define commands here \subsection{Introduction} Mathematics relies on precise symbolic language to express relationships between statements. Central to this language are \textbf{connective symbols}, also called \textbf{logical connectives}. These symbols combine simpler statements into more complex ones whose truth values depend in a well-defined way on their components. Logical connectives form the foundation of mathematical reasoning, formal proof, set theory, computer science, and many areas of physics and engineering. Every theorem, definition, and proof ultimately rests on combinations of these connectives. A statement that is either true or false is called a \textbf{proposition}. We typically denote propositions by lowercase letters such as $p$, $q$, and $r$. \subsection{Basic Logical Connectives} There are five primary logical connectives used throughout mathematics. \subsubsection{Negation} The \textbf{negation} of a proposition reverses its truth value. \[ \neg p \] This is read as \begin{quote} ``not $p$'' \end{quote} \textbf{Example:} If \[ p: \text{``The number is positive''} \] then \[ \neg p: \text{``The number is not positive''} \] \subsubsection{Conjunction} The \textbf{conjunction} of two propositions is true only if both are true. \[ p \land q \] This is read as \begin{quote} ``$p$ and $q$'' \end{quote} \textbf{Example:} \[ \text{``The number is positive and even''} \] \subsubsection{Disjunction} The \textbf{disjunction} represents logical ``or.'' \[ p \lor q \] This is the \textbf{inclusive or}, meaning one or both may be true. \textbf{Example:} \[ \text{``The number is negative or zero''} \] \subsubsection{Implication} The \textbf{implication} represents logical consequence. \[ p \rightarrow q \] This is read as \begin{quote} ``if $p$, then $q$'' \end{quote} Here: \begin{itemize} \item $p$ is called the \textbf{antecedent} \item $q$ is called the \textbf{consequent} \end{itemize} \textbf{Example:} \[ \text{If a number is divisible by 4, then it is even} \]