Question

hey i was wondering if anyone can tell me what are the types of theorems to proof and what they are and what they do please?

Answer 1

@Owlcoffee ?

Answer 2

Well there are not so many types of theorems.
We have existance theorems, like Bolzanos theorem but all the theorems have pretty much the same idea, they al have the same idea, a "hypothesis" and a "thesis".
Logically talking a theorem afirms a "provable truth".
Long ago, when mathematics took it's first steps, there were pretty much no theorems, because if we tried to prove, every single thing in mathematics, we sould still have no theorems.
So we then can take our another box, something we call "axioms", that go hand in hand with the "theorems".
In few words the axioms are "theorems" that have no proof, because we took them as true without any need of it.
And Theorems, they do have a rigurous proof, you can use an axiom or another theorem to write a rigurous proof.
We start of with an hypothesis (statements we hold true) and land on a thesis (conclusion).
we have "corolary", wich is a immediate consecuense of a theorems, wich we could prove from a theorem previously proven.

A theorem can also be a "well formed" formula that can be proven in a formal system, starting from axioms or other theorems. Proving a theorem is a thing of mathematical logic.
But there are also theorems in physics, chemistry, economy, etc.
Still they have the same idea, a rigurously proven formula or statement, that uses axioms or other theorems.

When a theorem is proven, it holds a truth under determined conditions. Like Rolles theorem.
But could also mean a property of some sort. Such like "Limit addition".
I also have recieved the question:
"Hey, but what if we have statement that we know is true, but havent proven?"
That is called a conjecture, a mathematical statement that we hold true but still not managed to prove. Like the shimura conjecture.

Theorems or conjectures can be proven in few ways but the most used of them is "contradiction", "direct", "mathematical induction", "construction" and the list goes on.