I apologize if this is an under specified question, but I will try and provide some context. I am beginning to learn the foundations of propositional and predicate logic after starting a course in abstract algebra (groups, fields, rings, etc).
I find that predicate logic comes very easy to me because of its relation to set theory...the quantifiers make the whole visualization of "which elements make a statement true" very intuitive and easy to work through. However, I am struggling to find a similar feature with propositional logic (perhaps because I have been trying to mimic my strategy from predicate logic).
I understand that one can use truth tables to attack propositional logic...but the truth tables seem very...artificial and not particularly insightful.
As an example, I commonly see this picture used to explain implication if $p$ then $q$ propositions:
Which, on its own, makes perfect sense. But I fail to see how this picture also represents $\neg p \lor q$, which is an equivalent form of $p \rightarrow q$. (As confirmed by truth tables)
If anything, the space "not $p$" would just be $q \land \neg p$...and therefore $\neg p \lor q$ would just be $(q \land \neg p) \lor q$...which is just $q$. But that means that $p \rightarrow q = q$...and that cannot be right, can it? That's sort of like modus ponens but missing the second step of decalring that $p$ is true.
Regardless, I'm clearly not thinking about this subject in an effective way so if anyone can offer insight, I'd greatly appreciate it!
Edit: I am adding some pictures to clarify some confusion:
Answers offered below have made me realize that the picture previously provided is incomplete. There should exist another set, call it $E$ (for Everything) that contains, in addition to q, everything outside of q. This changes my understanding of what $\neg p$ means...but questions nonetheless remain.
$\neg p$ and $q$ are therefore illustrated as follows:
Therefore, $\neg p \lor q$ looks like this:
To me this means that every element in the universe satisfies the proposition of $\neg p \lor q$...which by equivalence, means that every element in the universe satisfies the proposition $p \rightarrow q$. Could I have further clarification?