Summary
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. This is a method of direct proof. A proof by exhaustion typically contains two stages: A proof that the set of cases is exhaustive; i.e., that each instance of the statement to be proved matches the conditions of (at least) one of the cases. A proof of each of the cases. The prevalence of digital computers has greatly increased the convenience of using the method of exhaustion (e.g., the first computer-assisted proof of four color theorem in 1976), though such approaches can also be challenged on the basis of mathematical elegance. Expert systems can be used to arrive at answers to many of the questions posed to them. In theory, the proof by exhaustion method can be used whenever the number of cases is finite. However, because most mathematical sets are infinite, this method is rarely used to derive general mathematical results. In the Curry–Howard isomorphism, proof by exhaustion and case analysis are related to ML-style pattern matching. Proof by exhaustion can be used to prove that if an integer is a perfect cube, then it must be either a multiple of 9, 1 more than a multiple of 9, or 1 less than a multiple of 9. Proof: Each perfect cube is the cube of some integer n, where n is either a multiple of 3, 1 more than a multiple of 3, or 1 less than a multiple of 3. So these three cases are exhaustive: Case 1: If n = 3p, then n3 = 27p3, which is a multiple of 9. Case 2: If n = 3p + 1, then n3 = 27p3 + 27p2 + 9p + 1, which is 1 more than a multiple of 9. For instance, if n = 4 then n3 = 64 = 9×7 + 1. Case 3: If n = 3p − 1, then n3 = 27p3 − 27p2 + 9p − 1, which is 1 less than a multiple of 9. For instance, if n = 5 then n3 = 125 = 9×14 − 1. Q.E.D.
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Related concepts (2)
Mathematical proof
A mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation".
Mathematical induction
Mathematical induction is a method for proving that a statement is true for every natural number , that is, that the infinitely many cases all hold. Informal metaphors help to explain this technique, such as falling dominoes or climbing a ladder: Mathematical induction proves that we can climb as high as we like on a ladder, by proving that we can climb onto the bottom rung (the basis) and that from each rung we can climb up to the next one (the step). A proof by induction consists of two cases.
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