After separating valid conditional rules from invalid conditional reasoning, we now use the connectives $\land$ and $\lor$ as inference tools. Conjunction and disjunction rules explain when statements may be joined, separated, enlarged, or handled by cases. These rules appear constantly in proofs because many mathematical hypotheses are written with “and” or “or.” In this lesson, conjunction and disjunction rules will be stated, proved, and applied to short symbolic arguments.

Let $p$ and $q$ be statements. If $p$ is true and $q$ is true, then $p\land q$ is true: $$\begin{equation*} \begin{array}{c} p,\ q\\ \hline p\land q \end{array} \end{equation*}$$