Propositional Logic and Operators

Consider a different statement:

If it rains today, then we will stay home today.

This statement, which is written as "if $p$ then $q$", can also be equivalently restated as "We will stay home today if it rains today". (In this example, $p = \;($"$\text{it will rain today}$"$)$ and $q = \;($"$\text{we will stay home today}$"$)$.)

Note that a conditional statement of the form "if $p$ then $q$" is true either if $p$ is false (regardless of what $q$ is,) or if $q$ is true (regardless of what $p$ is.) In other words, the only case when this statement is false is when $p$ is true but $q$ is false (that is, when it indeed rains, but, despite this, we still go outside.)

This description of a conditional / implication statement tells us that we can express the proposition at the top as: $p \rightarrow q = (\neg p) \vee q = \neg p \vee q$ (not $p$, or $q$).