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In logic and mathematics, proof by example (sometimes known as inappropriate generalization) is a logical fallacy whereby the validity of a statement is illustrated through one or more examples or cases—rather than a full-fledged proof. [1] [2] The structure, argument form and formal form of a proof by example generally proceeds as follows ...
In proof by exhaustion, the conclusion is established by dividing it into a finite number of cases and proving each one separately. The number of cases sometimes can become very large. For example, the first proof of the four color theorem was a proof by exhaustion with 1,936 cases. This proof was controversial because the majority of the cases ...
In mathematics, a proof by infinite descent, also known as Fermat's method of descent, is a particular kind of proof by contradiction [1] used to show that a statement cannot possibly hold for any number, by showing that if the statement were to hold for a number, then the same would be true for a smaller number, leading to an infinite descent and ultimately a contradiction. [2]
A proof, we will see, is just that, a "test" of the theorem that we do by inserting a "proof example" into the beginning and see what pops out at the end. Both Lemmas #1 and #2 are required to form the necessary "IF AND ONLY IF" (i.e. logical equivalence) required by the proof:
Sometimes there are only two or three cases. Sometimes there may be thousands or even millions. For example, rigorously solving a chess endgame puzzle might involve considering a very large number of possible positions in the game tree of that problem. The first proof of the four colour theorem was a proof by exhaustion with 1834 cases. [4]
For example, suppose that after a while, the mathematician above settled on the new conjecture "All shapes that are rectangles and have four sides of equal length are squares". This conjecture has two parts to the hypothesis: the shape must be 'a rectangle' and must have 'four sides of equal length'.
This source of combinatorial explosion was eliminated in 1965 by John Alan Robinson's syntactical unification algorithm, which allowed one to instantiate the formula during the proof "on demand" just as far as needed to keep refutation completeness. [2] The clause produced by a resolution rule is sometimes called a resolvent.
This is generally considered one of the most important open questions in mathematics and theoretical computer science as it has far-reaching consequences to other problems in mathematics, to biology, [14] philosophy [15] and to cryptography (see P versus NP problem proof consequences). A common example of an NP problem not known to be in P is ...