Propositional Logic and Operators

Here is yet another statement:

We can drink coffee or tea and not eat any donuts, or just drink coffee.

This statement sounds quite odd and abstract, but we can still express it with our logic notation. If we denote $p = \;($"$\text{drink coffee}$"$)$, $q = \;($"$\text{drink tea}$"$)$, and $r = \;($"$\text{eat donuts}$"$)$, then the statement can be expressed as: $((p \vee q) \wedge (\neg r)) \vee p = \;$$(p \vee q) \wedge \neg r \vee p$.

We soon see that this proposition can be re-phrased is a shorter and simpler way. This is true for other long propositions, too: there might be a way of re-writing a proposition such that it uses fewer variables or instances of variables, but still fully retains its truth value.

Quick example: The proposition: "This coffee is great; this coffee is great; this coffee is great!", which we denote as $p \wedge p \wedge p$, can be re-phrased as simply "This coffee is great": $p$ (if we are referring to the same exact coffee sample.)