1.2 Techniques of Proof: Direct Proof

Objectives:

• Discover proof techniques for basic conditional propositions

• Apply methods of contrapositive and contradiction to several number theory examples

Example 1.2.1.

Reword the following so that it reads explicitly as a conditional proposition of the form "If , then ": “The sum of any three consecutive integers is always divisible by three.”

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We will attempt to prove this statement directly. A direct proof is a proof that assumes the antecedents of a conditional statements are true, then deduces from them that the consequences of the conditional statements are also true.

A Skeleton Version of a Direct Proof for $A\Rightarrow B$.

Guidelines	Example
Assume $A$.	Assume $n$, $m$, and $\mathrm{\ell }$ are three consecutive integers.
Use definitions to rewrite $A$. (There is sometimes more than one way to do this.)	Without loss of generality (WLOG) assume . Then $m=n+1$ and $\mathrm{\ell }=n+2$.
(Recommended) State what you wish to show in $B$ given what you’ve done so far.	We want to show that $n+m+\mathrm{\ell }$, or $n+(n+1)+(n+2)$, is divisible by 3.
Connect what you have to the definitions used in $B$.	???
Conclude $B$.	Then $n+m+\mathrm{\ell }$ is divisible by 3 by definition. This concludes the proof.

Example 1.2.2.

Fill in the blank box in the skeleton version for $A\Rightarrow B$ to complete the proof of the statement from Example 1.

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Boxes 2 and 4 are where the magic happens. Box 2 requires strong knowledge of definitions so you can apply them directly to the problem provided. Box 4 requires a connection between what you have and what you need.

The stronger one’s understanding of the definition, the more flexible you can be in starting your proofs off right. Perhaps this makes sense of the following quotes:

In mathematics, the art of proposing a question must be held of higher value than solving it.

- Georg Cantor

I kept an updated list of definitions/propositions/theorems/etc in order and went through their proofs the few days leading up to the exams so that they were fresh on my mind and make sure that I was able to do them on my own.

- A student from a previous proof-based course I taught

Example 1.2.3.

Prove the following theorem directly: “If $a,b,c\in \mathbb{Z}$ such that $a$ divides $b$ and $b$ divides $c$, then $a$ divides $c$.”

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(More space for Example 1.2.3.)

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Example 1.2.4.

If $m$ is a real number and $m$, $m+1$, and $m+2$ are the lengths of the three sides of a right triangle, then $m=3$.

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Example 1.2.5.

Disprove the following conjecture: “If $5$ divides $n^2-1$, then 5 divides $n-1$.”

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