Direct Proofs

Created at: 2025-05-10

A direct proof is a way to prove "P therefore Q", or P → Q.

The idea is to start with some P and work a way towards Q. This usually involves applying definitions, previous results, algebra, logic and other techniques to satisfy P → Q.

Structure

Proposition:
  P → Q

Proof:
  Assume P
  <explanation of what P means>
              ...
  Apply algebra, logic, etc.
              ...
  <this is what Q means>

Therefore Q

Example

Fact

• The sum, difference, and product of integers is an integer.
• An integer is either even or odd.

Definitions:

• An integer n is even if n = 2k for some integer k.
• An integer n is odd if n = 2k + 1 for some integer k.

Proposition:

• The sum of two even integers is even.

Given
  A = 2x
  B = 2y

Then
  A + B = 2x + 2y
  A + B = 2(x + y)

Let
  z = (x + y)

So that
  A + B = 2z

By definition:
  A + B is an even number.