Unlike the natural and social sciences, which largely rely on data, experiment, and statistical analyses, mathematical knowledge proceeds deductively via proof. But what is a proof? We will consider a proof to be a “convincing argument,” written as a short essay, and adhering to the standard rules of mathematical English. But what is an argument? And whom should it convince?
As an illustration, consider a statement of the form “If \(P\text{,}\) then \(Q\text{.}\)” A direct proof will begin with the assumption that statement \(P\) is true. We will use our knowledge of what it means for \(P\) to be true —using other definitions or theorems —to make a new deduction. We continue in this way until we may deduce that \(Q\) is true.
We now consider the following example proof. For the purposes of this proof and what follows, we may assume that the basic laws of arithemtic of integers are true: an integer times an integer is an integer, an integer plus an integer is an integer, etc.
Let \(m\) be an even integer. By Definition 3.1.1, there exists an integer \(k\) such that \(m = 2k\text{.}\) When we square \(m\text{,}\) we obtain \(m^2 = (2k)^2 = 2^2 \cdot k^2 = 4k^2\text{.}\) Observe that we may rewrite \(m^2\) as
To typeset an exponent in \(\LaTeX\text{,}\) we use the carat symbol, ^, in math mode. So, m^2 produces \(m^2\text{.}\) If the exponent has more than one character, enclose the expression in the exponent in curly braces; so, m^{n+2} produces \(m^{n+2}\text{.}\)
Note that we are not considering a specific even integer, such as 18, and verifying that its square is even. Part of the power of mathematics is that we have tools for proving general statements about all even integers at once. Proofs are not examples, and you shouldn’t treat them as such.
The proof avoids the use of pronouns to refer to a mathematical objects; we don’t square “it”, we square \(m\text{.}\) This keeps things clear and easy to follow.
Each step follows from the one before it, though specific reasons are not always cited. Determining when it’s appropriate to cite, say, a definition or theorem by number is a bit of an art form. We’ll figure it out.
You should consider your audience. If your audience is a group of research mathematicians, you’ll write a far different proof than if you are writing for your classmates. You should write for your classmates.
An axiom is an unproved assumption. Without axioms, we have no starting point for our explorations and deductions. However, since we do not prove our axioms, we should ensure that they are reasonable and clear. We will likely eventually prove Axiom 3.1.9, but for now we may take it as true without proof.
Let \(v\) be a vertex in a graph \(G\text{.}\) The degree of \(v\), denoted \(d(v)\text{,}\) is the number of edges incident to \(v\text{.}\) We say a vertex is even if its degree is an even integer, and odd if its degree is an odd integer.
Let \(G\) be a graph. The minimum degree of a vertex in \(G\) is denoted by \(\delta(G)\text{.}\) The maximum degree of a vertex in \(G\) is denoted by \(\Delta(G)\text{.}\)
