Methods of Proof

CS390, Fall 2019

Last modified: Aug 31, 2019
Contents:

Abstract

Chapter 1 surveys some common forms of mathematical proofs.

Here I present some examples of the forms we will use most often.

1 This is a Review!

The various forms of proof discussed in Chapter 1 of the text are not given in great detail. That’s because these are not supposed to be new to you.

For example, proof by induction, something that students seem to struggle with a lot, is according to the Virginia Standards of Learning, a pre-algebra (early high-school) concept. So, yes, you should be comfortable with the concept by now. No, it’s not covered in depth in the textbook, because that’s only intended as a quick refresher.

If you are really struggling with these methods of proof, you may need to do some review quickly. But don’t expect to find that in advanced texts or resources specific to automata theory and languages. Look for much more general mathematics resources instead. Some suggestions can be found on the Library page.

2 Examples: Methods of Proof

2.1 Deduction

2.2 Contradiction

2.3 Mathematical Induction

2.3.1 Believing in proof by induction

Think of proof by induction like setting up one of those tricks with dominoes where pushing over the first one in line causes all of the others to fall.

falling dominoes

For that sequence to work, two things have to happen.

  1. Someone must tip the first domino over.
  2. The rest of the dominoes must be positioned so that, whenever one falls, it strikes the one after it.

A proof by induction lines up an infinite sequence of mathematical problems of different “size”, n, where n can be any natural number.
We prove a theorem $P(n)$ by

  1. proving that it holds for $n=0$ (or sometimes $n=1$ or some other starting point, depending on the theorem) — someone has tipped the first domino, and

  2. the problems are “lined up” so that, if all the problems of size $0, 1, \ldots, k$ are true, then the problem of size $(k+1)$ will also be true — if the earlier dominoes have been tipped over, then the next one will tip over as well.

2.4 Structural Induction