Why isn't this a sufficient proof?

Question

So basically, we have a question that asks us to prove that given a particular Deterministic finite automaton (DFA), there is a symbol for which we can get to a state $q$ from a state $p$ given a function that takes $a$ and $q$ as arguments.

Can't we just show a particular instance for which that is true? For instance when a is equal to epsilon? What constitute a sufficient proof anyways? I thought by using contradiction for instance and showing it's not true or it is true for a particular case was sufficient.

Answer 1

"Can't we just show a particular instance for which that is true? For instance when a is equal to epsilon?"

1. No: a proof of a proposition, lemma, theorem, etc... must establish that all the claims make by it are true given the assumptions in the proposition, lemma, etc...

"What constitute a sufficient proof anyways? I thought by using contradiction for instance and showing it's not true or it is true for a particular case was sufficient."

1. To show a proposition, lemma, theorem, etc... is false, it is enough to exhibit one counter-example.So it is easier (in theory) to prove that something is false than it is true because to establish it is true (to prove it), you need to cover all possible cases.

2. A common (an old) technique of proving statements is "proof by contradiction" or reductio ad absurdum. There you create a new statement that says that your original statement (the one you want to prove) is false. Next you provide a counter-example to it. By doing so, you prove that the statement that says your original statement is false is false, which means your original statement is true.