Induction Definition and 999 Threads

Mathematical induction is a mathematical proof technique. It is essentially used to prove that a statement P(n) holds for every natural number n = 0, 1, 2, 3, . . . ; that is, the overall statement is a sequence of infinitely many cases P(0), P(1), P(2), P(3), . . . . 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. The first, the base case (or basis), proves the statement for n = 0 without assuming any knowledge of other cases. The second case, the induction step, proves that if the statement holds for any given case n = k, then it must also hold for the next case n = k + 1. These two steps establish that the statement holds for every natural number n. The base case does not necessarily begin with n = 0, but often with n = 1, and possibly with any fixed natural number n = N, establishing the truth of the statement for all natural numbers n ≥ N.
The method can be extended to prove statements about more general well-founded structures, such as trees; this generalization, known as structural induction, is used in mathematical logic and computer science. Mathematical induction in this extended sense is closely related to recursion. Mathematical induction is an inference rule used in formal proofs, and in some form is the foundation of all correctness proofs for computer programs.Although its name may suggest otherwise, mathematical induction should not be confused with inductive reasoning as used in philosophy (see Problem of induction). The mathematical method examines infinitely many cases to prove a general statement, but does so by a finite chain of deductive reasoning involving the variable n, which can take infinitely many values.

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  1. H

    What rules governs the mutual induction

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  2. A

    Stuck on Proof by induction of 2^n>n^3 for all n>=10

    Homework Statement Using the principle of mathematical induction, prove that for all n>=10, 2^n>n^3 Homework Equations 2^(n+1) = 2(2^n) (n+1)^3 = n^3 + 3n^2 + 3n +1 The Attempt at a Solution i) (Base case) Statement is true for n=10 ii)(inductive step) Suppose 2^n > n^3 for some integer >=...
  3. M

    MHB Induction on the number of equations

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  4. TheMathNoob

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  5. M

    Electromagnetic Induction in a moving DC circuit

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  6. K

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  7. G

    Proof by Induction: Proving Sum Equation for n ∈ N

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  8. A

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  9. T

    Proving 2n ≤ 2^n for All Positive Integers n

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  10. M

    Exploring Innovative Mathematical Strategies - Beyond Induction

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  11. Y

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  12. A

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  13. P

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  14. M

    Is Induction More Rigorous in First-Order Logic Than Higher-Order Logic?

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  15. C

    Proof using mathematical induction

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  16. TheMathNoob

    How Does Induction Prove the Probability Limit of Nested Sets?

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  17. C

    Engineering Where Does the Induction Motor Torque Formula Come From?

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  18. B

    What Happens to an Induction Generator at No Load and Super-Synchronous Speed?

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  19. S

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  20. C

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  21. BiGyElLoWhAt

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  22. C

    Finding the Magnetic Field in a System with Intersecting Current-Carrying Pieces

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  23. Viraj Thakkar

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  24. B

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  25. L

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  27. J

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  28. cnh1995

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  29. H Smith 94

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  30. K

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  31. D

    MHB How Does Coil Orientation Affect Induced EMF in Electromagnetic Induction?

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  32. Hyo X

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  33. D

    Hello Everyone....Single phase induction motor design

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  34. T

    Induction motor equations question

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  35. I

    Quantum Mechanics - Induction Method

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  36. PcumP_Ravenclaw

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  37. N

    Mutual Induction and Law of Conservation of Mass

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  38. TalkOrigin

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  39. TalkOrigin

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  40. V

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  41. J

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  42. Keen94

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  43. V

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  44. B

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  45. S

    How Many Turns are Required for an Induction Motor in Different Voltage Levels?

    I want to known the stator design of an indiction motor. How may turns required for each pole. For exmple if a single phase motors is designed to run at 230v ac, how may turns required in each pole. If the same motor is designed to be run in lower voltage (say 110v ac or 12v ac) what will be the...
  46. AlexiaIoannou

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  47. K

    Proving combination is a natural number by induction

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  48. phoenixthoth

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  49. PhysixChick

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  50. M

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