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: 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.


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

This is equivalent to saying:

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

We note the following mathematical conclusion:

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

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