Direct Proof

What is a Direct Proof?

The simplest kind of proof. You want to prove:

$P \to Q$

“If P, then Q.”

The Method

1. Assume P is true
2. Use logic to show Q must follow
3. Done

Start at P. Take logical steps. Arrive at Q.

Example 1: Even Squares

Claim: If $n$ is even, then $n^2$ is even.

Proof:

• Assume $n$ is even
• By definition, $n = 2k$ for some integer $k$

• This is $2 \times (\text{something})$, so $n^2$ is even

We assumed even, we got even. Done.

Example 2: Sum of Odds

Claim: The sum of two odd numbers is even.

Proof:

• Assume $a$ and $b$ are both odd
• By definition:
  • $a = 2m + 1$ for some integer $m$
  • $b = 2n + 1$ for some integer $n$

Add them together:

• This is $2 \times (\text{something})$, so the sum is even

Two odds always make an even.

Example 3: Consecutive Product

Claim: If $n$ is an integer, then $n(n+1)$ is even.

Proof:

• Assume $n$ is an integer
• Either $n$ is even or $n$ is odd

Case 1: $n$ is even

• $n(n+1) = \text{even} \times \text{anything} = \text{even}$

Case 2: $n$ is odd

• Then $n+1$ is even

• $n(n+1) = \text{anything} \times \text{even} = \text{even}$

• Either way, the product is even

Consecutive integers — one is always even.

The Pattern

Direct proofs often rely on definitions.

Plug in the definition, do the algebra, show what you need.

Definition in, result out.