Section 2.1: Proof Techniques

Abstract:

Sometimes we see patterns in nature and wonder if they hold in general: in such situations we are demonstrating inductive reasoning to propose a theorem, which we can attempt to prove via deductive reasoning. From our work in Chapter 1, we conceive of a theorem as an argument of the form , which we seek to demonstrate is valid.

This section introduces us to a variety of proof techniques, including direct proofs, proofs by contraposition, proofs by contradiction, proofs by exhaustion, and proofs by dumb luck or genius!

Theorems and Informal Proofs

The theorem-forming process is one in which we

• make observations about nature, about a system under study, etc.;
• discover patterns which appear to hold in general;
• state the rule; and then
• attempt to prove it, or disprove it.

This process is formalized in the following definitions:

• inductive reasoning - drawing a conclusion based on experience
• deductive reasoning - application of a logic system to investigate a proposed conclusion based on hypotheses (hence proving, disproving, or, failing either, holding in limbo the conclusion).
• counterexample - an example which violates a proposed rule (or theorem), proving that the rule doesn't work in general.

Before attempting to prove a theorem, we should be convinced of its correctness; if we doubt it, then we should pursue the line of our doubt, and attempt to find a counterexample.

Kids are wonderful at developing conjectures, and sometimes even applying deductive logic (Sam's Story). Practice 1, p. 85 illustrates the kinds of conjectures kids will make (e.g. ``All animals living in the ocean are fish.''), and parents, sibling, friend, and teachers all have the priviledge and pleasure of coming up with counterexamples.

Exhaustive Proof

The Four-color problem

• Description (see p. 361).
• This theorem is partly famous because it provided the first example of a computer-aided proof of a major result. The reason the computer became useful was that the proof came down to testing a rather large number of special cases.

When there are only a few things (in particular, a finite number) to test, we can use proof by exhaustion.

Example: Prolog is able to test conjections, or theorems, such as in-food-chain(bear,algae) by simply doing a proof by exhaustion: it checks all cases, and eventually finds that algae is indeed in the bear's food chain.

Direct Proof

The most obvious, and perhaps common technique, is the direct proof: you start with your hypotheses , and proceed toward your conclusion Q:

Example: Exercise 12, p. 93

Contraposition

If isn't getting you anywhere, you can use your logic systems to rewrite it as (the contrapositive). This is called ``proof by contraposition''.

Example: Practice 4, p. 89

Example: Exercise 15, p. 93

Contradiction

Contradiction represents some interesting logic: again, we want to prove , but rather than proceed directly, we seek to demonstrate that : that is, that P and Q' leads to a contradiction. Then we cannot have both P true, and Q false - which would lead to false, of course.

Example: Exercise 19, p. 93

  
Table: Summary of useful proof techniques, from Gersting, p. 91.

Serendipity

Mathematicians often spend a great deal of time finding the most ``elegant'' proof of a theorem, or the shortest proof, or the most intuitive proof. We may stumble across a beautiful proof quite by accident (``serendipitously''), and those are perhaps the most pleasant proofs of all. There is a wonderful story associated with Exercise 50, p. 95.