When someone talks about "Theorems" in Mathematics, something of the sort below comes to my mind.
Theorem 1: For every two real numbers $a$ and $b$ with $a \lt b$, there exists a rational number $r$ satisfying $a \lt r \lt b$.
Basically, there's a Hypothesis (the "if" part) and a Conclusion (the "then" part). So if I were to, say, prove this theorem using contradiction, I would start by negating the conclusion and then proceeding logically until a contradiction is found with the hypothesis (contrapositive proof) or some other accepted fact. The hypothesis and conclusion can be seen very easily here. Let us take a different example.
Theorem 2: There exists no rational number $r$ whose square is $2$.
I am having a hard time seeing what the hypothesis is in this case. Maybe if this theorem is worded in a different way, it might be obvious but that's just my opinion.
And if you're familiar with the proof of this theorem, the contradiction comes from the fact that we had supposed a rational number $\frac pq$ in lowest terms but after proceeding logically, we find a factor of $2$ common between $p$ and $q$ which goes against our assumption that we started with and so our proof ends.
So my questions are:
- Are all "Theorems" necessarily in the "if-then" form?
- What is the Hypothesis in Theorem $2$? Is the proof for this theorem a contrapositive proof?
I heartily welcome any extra information in relation to this question.