Direct Proof

A direct proof invloves taking a proposition and proving it true. Say we had the statment "If it is cold then it is raining." in a direct proof we would start by assuming it is cold and then finding a way to show how that means its raining. In the study of logic they call this Modus Ponens.

Example:

Let n be an integer. Prove that if n is divisible by 21, then n is divisible by 3.

Proof:

Let n be some integer that is divisible by 21. By Definition 1, let k be an integer that satisfies n=21*k. Consider

n = 21*k
n = 3*(7*k)

We know that 7*k is an integer since 7 and k are both integers (Axiom 3). Thus, by Definition 1, we can say n is divisible by 3 since there exists an integer i such that n=3*i where i=7*k. We have shown that if n is divisible by 21, then n is divisible by 3.