We prove a lot of statements by contradiction.
- We got a statement.
- Then suppose its negation.
- Derive a contradiction
But what if instead of deriving contradiction I just show a case where the negation fails to be true. Will it mean that the statement is true? Because it seems to me like a cheating. For motivational example suppose our statement is "Square of an even number is also even".
We could prove it by contradiction, showing the square of $2m$ is divisible by $2$ that is why it is a contradiction.
But what about showing a simple case to disprove the negation?
Suppose it is false, i.e. the square number is odd. Let $a=2$. Then square is $2^2 = 4$. But 4 is not odd, since it is divisible by 2. Then the statement is true.
It is like thinking of the negation as a new statement. And we can disprove any statement by showing a case where it fails to be true. Is it acceptable? As I said, I think of this more as a cheating, because showing a contradiction is more general unlike showing a case.