Axiom 3.0.1.
Let \(n\) and \(m\) be integers. Then the following numbers are also integers
\begin{align*}
-n, \amp\amp n+m, \amp\amp n-m \amp\amp\text{ and } \amp\amp nm.
\end{align*}
Chapter 3 Direct proofs
Before we get to actually proving things we should spend a little time looking at how we name and prioritise mathematical statements. Not all things we want to prove are created equal and as a consequence they get different names.
- Axioms
- Axioms are these are statements we accept as true without proof. Clearly they are very important since all our work hangs on them.
- Facts
- We can also state some things as facts — these might be provable from axioms, but for the purposes of our text we don’t want to go to the trouble (effort?) of proving them.
In this text we will use the following as an axiom.
The authors are going to assume that you are familiar with the above properties and we do not need to delve deeper into them. The following is a statement that can be proved from the standard axioms of real numbers — we are not going to prove that, but we will use it. So we’ll state it as a fact.
Fact 3.0.2.
Let \(x\) be a real number. Then \(x^2 \geq 0\text{.}\)
Another useful fact is Euclidean Divison, also called the division algorithm by some texts. It will come in very handy when we discuss even and odd numbers (for example).
Fact 3.0.3. Euclidean division.
Given integers \(a,b\) with \(b>0\text{,}\) we can always find unique integers \(q,r\) so that
\begin{equation*}
a = bq+r
\end{equation*}
with \(0\leq r < b\text{.}\)
So axioms and facts are slightly odd in that we don’t have to prove them, but lets move onto statements that we do prove to be true.
- Theorems
- A Theorem is a true statement that is important and interesting — Pythagorous’ theorem for example. Or Euclid’s theorem stating that there are an infinite number of prime numbers. Also, it is sometimes the case that implicit in the use of the word “Theorem” is that this is a result that we will use later to build other interesting results.
- Corollary
- A Corollary is a true statement that is a consequence of a previous theorem. Of course, this makes almost everything a corollary of something else, but we tend to only use the term when the corollary is a useful (and fairly immediate?) consequence of a theorem.
- Lemma
- A Lemma is a true statement that by itself might not be so interesting, but will help us build a more important result (such as a theorem). It is a helping result or a stepping stone to a bigger result 33 . You will occasionally see lemma pluralised as “lemmata”.
- Result and Proposition
- Otherwise we might just call a true statement a “Result” (especially if it is just an exercise or an example) or perhaps, if a little more important, a “Proposition”.
Indeed, the German word for Lemma is “Hilfssatz” — a helping result
