Mathematical induction is a method for proving that a statement P is true for every natural number n, that is, that the infinitely many cases P, P, P, P,.
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13 Facts About Mathematical induction
2.
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 and that from each rung we can climb up to the next one .
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3.
Mathematical induction is an inference rule used in formal proofs, and is the foundation of most correctness proofs for computer programs.
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4.
The first explicit formulation of the principle of Mathematical induction was given by Pascal in his Traite du triangle arithmetique .
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5.
Conclusion: Since both the base case and the induction step have been proved as true, by mathematical induction the statement P holds for every natural number n ?.
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6.
In practice, proofs by Mathematical induction are often structured differently, depending on the exact nature of the property to be proven.
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7.
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.
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8.
Complete Mathematical induction is most useful when several instances of the inductive hypothesis are required for each Mathematical induction step.
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9.
The Mathematical induction hypothesis applies to and, so each one is a product of primes.
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10.
However, the argument used in the Mathematical induction step is incorrect for, because the statement that "the two sets overlap" is false for and.
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11.
The axiom of Mathematical induction asserts the validity of inferring that holds for any natural number from the base case and the Mathematical induction step.
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12.
Strictly speaking, it is not necessary in transfinite induction to prove a base case, because it is a vacuous special case of the proposition that if P is true of all, then P is true of m It is vacuously true precisely because there are no values of that could serve as counterexamples.
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13.
Principle of mathematical induction is usually stated as an axiom of the natural numbers; see Peano axioms.
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