Proof

Definition: Suppose are all the axioms and previously proved theorems of a mathematical system. A formal proof , or deduction, of a sentence P is a sequence of statements , where

1. is P, and one of the following holds
2. 1. is one of , or
   2. follows form the previous statements by a valid argument using the rules of reasoning.

A theorem  is any sentence deduced from the axioms and/or the previous theorems. The same is true of lemma and proposition. For some mathematicians there is a hierarchy of lemma, proposition, and theorem; with lemma being the easiest to prove and theorem the most difficult, or longest. Other mathematicians make little or no distinction between these objects, and will call everything a theorem.

Example: Suppose a mathematical system contains just the following axioms:

:

:

a+b=c.

The following is a formal proof of x<y.

In practice mathematicians do not write formal proofs. They write informal proofs. An informal proof is an argument which shows the existence of a formal proof. As such it gives enough of the formal proof so that another person becomes convinced. Thus we might call an informal proof a convincing argument. Mathematicians try to convince other mathematicians. You will try to convince your fellow students and me, your professor.

An informal proof of the above example runs as follows:

From and it follows that . Thus, x<y.

Henceforth, we will be writing only informal proofs. The art of mathematics is creating proofs. Just as every other artisan, the mathematician has some basic modes of proof. We will now consider a few of these.

 
• Proving Conditionals
• Proving Biconditionals
• Proving
• Proof by Cases
• Mathematical Induction
• Proof by Contradiction
• Proofs of Existence and Uniqueness