How do you prove a statement is true “in general”
Is it enough to prove that the statement is true if and only if it meets a specific condition? Because that would prove the statement is false for anything that doesn’t meet second condition? Almost like an inverse counter-example. It’s only true in this one specific scenario, so it cannot be true in general
For example (not an accurate one, idr the specific wording, but I hope it demonstrates what I mean):
Let f: X -> Y and A sub B sub X. Prove that, in general, f(B) not sub f(A)
Prove f(A) sub f(B)
Therefore, if f(B) sub f(A), f(B) = f(A)
Prove that f(B)= f(A) iff f is injective
Injective is a specific case, so the statement cannot be true in general
Not interested in the validity of this particular proof itself so much as whether or not the logic is sound. Is there anything I would need to add for the conclusion?