I am a high school student interested in propositional logic. I have been studying many examples and decided to attempt to write my own. I feel like it's very badly done and would appreciate some feedback. (This is the first propositional proof I have EVER written, I came up with the problem myself and I am aware it's not original and very basic.)
Prove that $$P+Q+R = P+(Q+R)$$
My Proof:
$1.$ $\mathrm Let$ $$P=7, Q=8, R=9$$ $2.$ $$P+Q+R=24$$ $3.$ $$Q+R=17$$ $$So$$ $$P+(Q+R) = 24$$ $4.$ $$Therefore$$ $$P+Q+R = P+(Q+R)$$
This is wrong. When of comes to logic, showing a specific case is not enough to prove something. Showing many cases is not enough either. To prove of you have to either show it's a direct consequence of definition or a consequence of something that was already proven. There are various proof techniques you can use. You could probably start working your way from the definition of addition as the recursive application of the successor function S(n).