Right, hi everyone. Sometime ago in the not too distant past, I downloaded a short paper titled 'Basic Proof Techniques' written by David Ferry (I will drop the link below). Long story short, Ferry goes over 4-types of proof: Direct Proof, Proof by Contradiction, Proof by Induction and Proof by Contrapositive. In order to demonstrate how each proof unfolds, he uses the theorem: If a and b are consecutive integers, then the sum of a + b is odd. Ok, fine.
Now's a splendid time to announce that math and I go together like England and the Euro's go together…we don't. That said, here is his direct proof of the theorem:
'Proof. Assume that a and b are consecutive integers. Because a and b are
consecutive we know that b = a + 1. Thus, the sum a + b may be re-written as
2a + 1. Thus, there exists a number k such that a + b = 2k + 1 so the sum a + b
is odd.'
I'm having trouble wrapping my head around the bolded sentence (I bolded it for emphasis). For example, let's say a = 2 and b = 3 (as 2,3 are consecutive integers) 3 = 2 + 1. So, 2 (a) + 3 (b) = 5 and 2(2) + 1 = 5 also. BUT HOW??? WHERE AND WHY DOES THE '2' COME INTO PLAY?!?!?!? Perhaps this question is too vague to be given a precise answer, but it has given me an absolute massive headache (one too massive to not be asked on this forum).
I must depart now for food, so thanks in advance for everyone's help…appreciate it greatly I do!
Edit:
Here is the Link: https://www.cse.wustl.edu/~cytron/547Pages/f14/IntroToProofs_Final.pdf
Best Answer
$2 + 3 = 2+(2 + 1) = (2 + 2) + 1 = 2\times 2 + 1$.
And $17 + 18 = 17 + (17 + 1) = (17+17) + 1 = 2\times 17 + 1$.
And if $a = a$ and $b = a+1$ then
$a + b = a + (a+1) = (a+a) + 1 = 2a + 1$.