10.2 Definitions, Theorems, Proofs

Mathematical writing is built from a few basic parts. The most important are definitions, theorems, and proofs. A definition explains what a word or object means. A theorem states something true. A proof explains why it is true.

These parts work together. A theorem uses words that must already be defined. A proof uses definitions and known facts to justify the theorem. If the definitions are unclear, the theorem becomes unclear. If the theorem is unclear, the proof has no clear target.

Definitions

A definition introduces meaning. It tells the reader exactly how a term will be used.

For example:

An integer $n$ is even if there exists an integer $k$ such that

This definition gives a precise test. The number $10$ is even because $10 = 2 \times 5$. The number $7$ is not even because there is no integer $k$ such that $7 = 2k$.

A good definition should be clear, usable, and placed before it is needed. The reader should not have to guess what an important word means.

Theorems

A theorem states a result that has been proved or will be proved.

For example:

Theorem. The sum of two even integers is even.

This statement has two parts. The assumption is that the two integers are even. The conclusion is that their sum is even.

A theorem should say exactly what is assumed and exactly what is claimed. Vague statements are hard to prove because the target is unclear.

Proofs

A proof explains why a theorem is true. It should connect the assumptions to the conclusion by clear steps.

For the theorem above, the proof can use the definition of even.

Let $a$ and $b$ be even integers. By definition, there exist integers $m$ and $n$ such that

and

Then

Since $m+n$ is an integer, $a+b$ is equal to $2$ times an integer. Therefore, $a+b$ is even.

This proof works because each step has a reason. We used the definition of even, basic algebra, and the fact that the sum of two integers is an integer.

How they fit together

Definitions give meaning. Theorems make claims. Proofs justify those claims.

For example:

Definition: even means divisible by $2$.

Theorem: the sum of two even integers is even.

Proof: write each even integer as $2$ times another integer, then factor out $2$.

This pattern appears throughout mathematics. Before proving something, we make sure the words are defined. Before writing a proof, we make sure the claim is stated clearly.

Common mistakes

A common mistake is to use a term before defining it. Another mistake is to state a theorem too vaguely. For example, “even numbers behave well under addition” is not a good theorem. It suggests an idea, but it does not state a precise claim.

A better theorem is:

Theorem. If $a$ and $b$ are even integers, then $a+b$ is even.

Another common mistake is to write a proof as a list of symbols with no explanation. Symbols are useful, but the reader also needs words that explain why each step follows.

A practical habit

When writing mathematics, ask three questions.

What terms must be defined? What exactly is being claimed? Why does the claim follow?

These questions keep the writing clear. They also help the reader see the structure of the argument.