• Structure of a Direct Proof
• Examples

Structure of a Direct Proof

A direct proof is a type of proof that you are probably most familiar with. The basic structure of a direct proof is a single implication as follows: $P \Rightarrow Q$ where P and Q are the hypothesis and conclusion, respectively. For a direct proof, you want to assume P and through logic, prove that Q is also true. Starting with P, you form a series of implications that eventually ends up concluding that Q is true. Let’s walk through an example.

Examples

Theorem: Every odd integer is the difference of two perfect squares.

This is equivalent to saying:

$$(\forall x \in \mathbb{Z})(\text{x is odd} \Rightarrow \exists a \exists b \in \mathbb{Z} \text{ such that } a^2 - b^2 = x)$$

Proof: By definition, if we let $c = 2b + 1$ where $b$ in an integer, then $c$ must be odd. We want to prove that $c$ can be made up of the difference between two perfect squares.

We note the following mathematical conclusion:

$$\begin{align*} (b + 1)^2 - b^2 &= b^2 + 2b + 1 - b^2 \\ &= 2b + 1 \\ &= c \end{align*}$$

As you can see from our work above, we made a chain of implications, (if this, then that) statements to eventually conclude that $c = 2b + 1$ which is what we assumed at the beginning. This is the basis of a direct proof.

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