Mathematical Proof/Methods of Proof/Other Proof Types

From Wikibooks, open books for an open world
Jump to navigation Jump to search

Other kinds of Proof[edit | edit source]

Some Proofs do not fall into any of the categories listed above. For example, a non constructive existence proof is a method which demonstrates the existence of a mathematical entity, without actually constructing it.

Non Constructive Existence Proofs[edit | edit source]

For example, proofs exist which show the existence of nonconstructible numbers (numbers which can not be created using any combination of algebraic operations). These proofs can clearly not be constructive.

Proof By Exhaustion[edit | edit source]

Proofs which analyse and document every instance of an assertion in a case-wise analysis can be used in finite cases. This is a cumbersome method of proof and is really only suitable when dealing with fairly small sets. A more involved use of proof by exhaustion is the proof of the Four-Colour Theorem. The proof has broken down the four-colour problem into subsets and a four-colour proof was applied to each of these by a computer. This proof has not currently been verified by a human and is hence not considered fully proven.