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Proof by exhaustion, also known as proof by cases, proof by case analysis, complete induction or the brute force method, is a method of mathematical proof in which the statement to be proved is split into a finite number of cases or sets of equivalent cases, and where each type of case is checked to see if the proposition in question holds. [1]
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 propositional logic, disjunction elimination [1] [2] (sometimes named proof by cases, case analysis, or or elimination) is the valid argument form and rule of inference that allows one to eliminate a disjunctive statement from a logical proof.
Complete induction is equivalent to ordinary mathematical induction as described above, in the sense that a proof by one method can be transformed into a proof by the other. Suppose there is a proof of () by complete induction. Then, this proof can be transformed into an ordinary induction proof by assuming a stronger inductive hypothesis.
Carter & Wagon (1994a) provide an alternative proof by dissection. They show how to partition the sectors into smaller pieces so that each piece in an odd-numbered sector has a congruent piece in an even-numbered sector, and vice versa. Frederickson (2012) gave a family of dissection proofs for all cases (in which the number of sectors is 8, 12 ...
This division into cases not only indicates which sequences are more difficult to handle, but it also reveals the important role denseness plays in the proof. [proof 1] In the first case, P is not dense in [a, b]. By definition, P is dense in [a, b] if and only if for all subintervals (c, d) of [a, b], there is an x ∈ P such that x ∈ (c, d).
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Proof systems in propositional logic can be broadly classified into semantic proof systems and syntactic proof systems, [86] [87] [88] according to the kind of logical consequence that they rely on: semantic proof systems rely on semantic consequence (), [89] whereas syntactic proof systems rely on syntactic consequence (). [90]