6.1 Direct proof
A direct proof is probably what you intuitively think of when asked to “prove” something.
Often we try to prove statements that are true for some cases - for example
Show that if is an odd positive integer, then is also odd.
We start by assuming that is an odd positive integer. To do this, we want to find an algebraic way to represent . As every odd number can be written in the form11 1 This can be proved by induction (see Section 6.3) for suitable (e.g. , , etc.) we can write as . From there we apply our assumption to show that the result is true
As is in the form22 2 It’s not the specific expression or variable names which we chose that are important here - it’s the overarching structure of the expression - i.e. that it’s in the form which is important. , we have shown that this is true.