Proofs

In mathematics every formula, equation, and theorem must be proven before it can be used. In other words before we use anything we must make a logical argument that it is true. In many cases this has already been done, allowing us to just use formulas whenever and how ever we want.

There are many ways to prove a proposition. All of which have a framework based on 3 basic structures: Directs Proofs, Proof by Cases, Proof by Contradiction, and Proof by Induction.

Many proofs will reference theorems, definitions, and axioms without proving them to be true. Generally they list everything they will use before beginning a proof. Referencing statements that have already been proven before is a great way to shorten a proof. Here is a list of assumed propositions that will be referenced on this site. We generally do not reference the use axioms but theorems and definitions will be referenced.

Axiom 1:

Arithmetic (All basic operations +, -, *, /)

Axiom 2:

Commutative, Associative, and Distributive properties.

Axiom 3:

Closure of integers under addition and multiplication. (After two integers are operated by addition or multiplication the result is still and integer.)

Theorem 1:

Let m,n be integers.

1. If m,n are even then m+n and m*n are even.
2. If m,n are odd then m*n is odd and m+n is even.
3. If m is odd and n is even then m*n is even and m+n is odd.

Definition 1:

Divisibility. Let a,b be integers with a≠0. We say b is divisible by a, if there exists an integer k such that b=k*a.