Direct Proof

A direct proof of a mathematical statement is a valid argument, in words, that shows a mathematical statement is tautologically true given a set of assumptions.

Example

Claim: An odd plus an odd is an even.

Proof: One must be very precise with definitions when doing a mathematical proof. In this case, by an odd we mean an integer in the set $\{1, -1, 3, -3, 5, -5, \dots\}$ and by an even we mean an integer in the set $\{0, 2, -2, 4, -4, \dots\}.$

By a quick check, you can see that sometimes and odd plus an odd is an odd. For example, $1$ and $3$ are both odds, and $1+3 = 4$ is even. Similarly, $-5$ and $5$ are both odd, and $-5 + 5 = 0$ is an even. However, it is impossible to check all cases by hand. To show the statement is always true, we need a proof.

Another way to write the set of odds is $\{x : x = 2k+1 \text{ for some integer }k\},$ and another way to write the set of evens is $\{x : x = 2k \text{ for some integer }k\}.$ For example, $5$ is odd and $5 = 2 \cdot 2 + 1.$ Similarly, $-7$ is odd and $-7 = -4 \cdot 2 + 1.$ On the other hand, $6$ is even and $6 = 3 \cdot 2,$ and $0$ is even and $0 = 0 \cdot 2.$

Let $a$ and $b$ be odd numbers. Then there exists integers $c$ and $d$ such that $a = 2c+1$ and $b = 2d+1.$ So, $$\begin{align} a+b & = (2c+1)+(2d+1) \\ & = 2c + 2d + 2 \\ & = (c + d + 1) \cdot 2 \end{align}$$ Since $c+d+1$ is an integer, $(c+d+1) \cdot 2$ is an even. Since $a$ and $b$ could be any odd numbers, any odd plus any odd is an even.

Proofs

In general, a proof will have the structure above. There will first be a claim, which will be called a Claim, Lemma, Corollary or Theorem. Then there will be the start of a argument, which will be either Proof or Counter Example.