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Mathematical induction is an inference rule used in formal proofs, and is the foundation of most correctness proofs for computer programs. [ 3 ] Despite its name, mathematical induction differs fundamentally from inductive reasoning as used in philosophy , in which the examination of many cases results in a probable conclusion.
All horses are the same color is a falsidical paradox that arises from a flawed use of mathematical induction to prove the statement All horses are the same color. [1] There is no actual contradiction, as these arguments have a crucial flaw that makes them incorrect.
In proof by mathematical induction, a single "base case" is proved, and an "induction rule" is proved that establishes that any arbitrary case implies the next case. Since in principle the induction rule can be applied repeatedly (starting from the proved base case), it follows that all (usually infinitely many) cases are provable. [ 15 ]
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]
Fermat's little theorem and some proofs; Gödel's completeness theorem and its original proof; Mathematical induction and a proof; Proof that 0.999... equals 1; Proof that 22/7 exceeds π; Proof that e is irrational; Proof that π is irrational; Proof that the sum of the reciprocals of the primes diverges
Another method of proof is by mathematical induction: We loosen the condition p > q {\displaystyle p>q} to p ≥ q {\displaystyle p\geq q} . Clearly, the theorem is correct when p = q {\displaystyle p=q} , since in this case the first candidate will not be strictly ahead after all the votes have been counted (so the probability is 0).
2.2 Inductive and algebraic proofs. 2.2.1 Inductive proof. 2.2.2 Algebraic proof. ... This identity can be proven by mathematical induction on . Base case Let ...
The proof of the Miech's theorem uses Brun sieve. If there is a hypothetical probabilistic density sieve , using the Miech's theorem can prove the Schinzel's hypothesis H in all cases by mathematical induction .