Mathematical Proof/Methods of Proof/Constructive Proof A
constructive proof is the most basic kind of proof there is. It
is a proof that starts with a hypothesis, and a person uses a
series of logical steps and a list of axioms, to arrive at a
conclusion. Parts of a Theorem A Theorem is a proven statement
that was constructed using an axiom. Some theorems are very
complicated and involved, so we will discuss their different
parts. The hypothesis is the "if" statement of a theorem. In a
way, it is similar to an axiom because it is assumed to be true
in order to prove a theorem. We will consider a simple example.
- Theorem 2.1.1. If A and B are sets such that and , then A=B.
In this theorem, the hypothesis is everything before the word
"then." This is a very simple proof. We need to prove that for
every x, . For the purpose of analyzing proofs, we will define
and . The most common way to prove an if and only if statement
is to prove necessity and sufficiency separately. So we start by
showing that Therefore, we