1.2 Some Useful Laws
In the previous section, we combined statements via logical operations. There is nothing stopping us combining these new statements too! We just need to know how to do it systematically. This relies on distributive laws and De Morgan’s laws. Here, \(P,Q,R\) are statements.
\[\begin{align*} P \wedge (Q \; \vee R) &\Leftrightarrow (P \wedge Q)\; \vee (P \wedge R),\\ P \; \vee (Q \wedge R) &\Leftrightarrow (P \; \vee Q) \wedge (P \, \vee R). \end{align*}\]
\[\begin{align*} \neg (P \, \vee Q) &\Leftrightarrow (\neg P) \wedge (\neg Q),\\ \neg (P \wedge Q) &\Leftrightarrow (\neg P) \, \vee (\neg Q). \end{align*}\]
If you’re ever in a situation where you need to prove these laws, use truth tables! They’re also really helpful when writing complex statements in simpler forms (see, for example, Homework Questions 2 & 3).