What is Induction: Definition and 999 Discussions

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.

View More On Wikipedia.org
Recent contents Top users
  1. A

    Is Induction Proof Needed for this Limit Problem?

    Prove that for fixed s > 0, we have lim e^-sN ( s sin bN + b cos bN ) = 0 N -> This problem was in a chapter on Laplace Transforms. I'm assuming this will require proof by induction.
  2. J

    Electromagnetic induction question

    I seem to have a fundamental misunderstanding of the principle here. in the previous part it was required to sketch the graph for the hall probe. I knew since the speed was constant the field was cut at a constant rate hence a straight line was needed. However I did not understand why it...
  3. R

    Electromagnetic induction of a wire in a complete circuit

    Say for you had a wire in a complete circuit inside a magnetic field (pointing inwards) perpendicular to the wire. You move the wire across (to the right) , cutting lines of flux, this induces a current in the wire. The induced current acts upwards using the dynamo rule (thumb is motion...
  4. ranju

    Finding Full Load Current & Torque in Induction Motors

    In an induction motor problem , I need to find the full load current & torque..for that I calculated the impedance with given data..& I got the impedance in polar form..But in the solution for finding the full load current they put only the magnitude of the impedance & not the phase..! I didn't...
  5. N

    What is the working principle of an isolated induction generator?

    can anyone give me a detailed description of working of isolated induction generator or self excited induction generator it has been mentioned about it in the following link in the last paragraph but no details are given...
  6. N

    Poly-phase Induction generator

    What will happen if the rotor is made to rotate at double the synchronous speed i.e slip becomes less than -1. I have never seen any diagram showing torque slip curve with slip less than -1.
  7. J

    Magnetic field and electric field induce one another forever

    A changing electric field produces magnetic field and vice versa.Does that mean that this process will carry on forever?Think of a circuit with a capacitor.The magnetic field due to the current at a point on the wire(with Ampere-Maxwell law).But current was changing with time,so it also meant...
  8. J

    Electromagnetic induction of a closed circular wire

    Homework Statement (See image) a closed circular wire A lies in the plane of the wire B in the circuit. Deduce the direction. Of the induced current in A when (A) the switch is closed (B) the switch is opened (C) the wire A is raised out of the plane of the paper. Homework EquationsThe...
  9. C

    Mathematical induction example

    Homework Statement A step in this process of proving Sn: 1+4+7+...+(3n-2) = n(3n-1)/2 confuses me. I hope someone can clarify this for me. I do not require the work done, I need clarification on a step only. Thanks! Homework Equations After assuming n=k, we say Sk: 1+4+7+...+(3k-2) =...
  10. D

    Homework Help on Kirchoff's Laws and Magnetic Induction

    Homework Statement I am stuck on 2 and 3. I attached all the problems and variable and unknownsHomework Equations I attached my work for 1, 2, and 3. I'm not sure if it is right though. [/B]The Attempt at a Solution Can someone check my work for problems 1, 2, 3 and 4? The equations...
  11. N

    Why crawling happens in induction motor?

    It is said that crawling occurs mainly due to 7th harmonic component. Due to 7th harmonic the speed decreases by the ratio of 7. However speed is directly proportional to frequency. So speed should get multiplied by a factor of 7 instead. Should the change in frequency affect the number of poles...
  12. S

    Introductory mathematical induction problem

    Homework Statement I am just learning the joys of mathematical induction, and this problem is giving me fits.Homework Equations I am trying to prove that 2 + 4 + 6 + … + 2n = [2n(n+1)]/2The Attempt at a Solution The base case is to prove P(1) is correct. Simple enough -- 2 = [2 x 1 (1+1)]/2...
  13. moriheru

    Proving ∑mk=1k2=1/6(323+3m2+m) by Induction

    ∑mk=1k2=1[/SUB]=1/6(323+3m2+m) How can I prove this by induction (m+1...) Prove for example of m, substitute m+1 into the equation, find the sigma K2 and solve the equation?
  14. Pami

    Prove gamma (n+1/2) = (2npi^1/2)/(n4^n) by induction

    I tried solving this question this way: Gamma(n+1/2) =(n+1/2-1)gamma(n+1/2-1) =(n-1/2)gamma(n-1/2) =(2n-1)/2 gamma (2 n-1)/2 Don't know what to do next
  15. A

    Electromagnetic induction of two wires

    I have two wires in parallel at a distance r apart. They are unequal in length. The current carrying primary generates a magnetic field B at distance r. What is the area related to the secondary through which B passes? Is it correct to assume the thickness times the length of the secondary? All...
  16. 1

    Proof by induction: n^3 < n for n >=6

    Homework Statement Show that n^3 < n! for all n >= 6. Homework EquationsThe Attempt at a Solution We see that for the base case of n = 6, the claim holds. Suppose that k^3 < k! for some natural number k >= 6. Consider that: (k+1)^3 = k^3 + 3k^2 + 3k + 1 < k! + 3k^2 + 3k + 1 [By induction...
  17. Jackson Lee

    A question about electromagnetic induction

    Hey, guys, I encountered this and confused about it. "In a perfect conductor, an arbitrarily large current can be induced, and the resulting magnetic field exactly cancels the applied field." But I suppose, according to Faraday's law,the magnitude of emf is the rate of change of magnetic flux...
  18. ellipsis

    Prove recurrence relation via mathematical induction

    $$ T(n) = \begin{cases} 2 & \text{if } n = 2 \\ 2T(\frac{n}{2})+n & \text{if } n = 2^k \text{, for } k > 1 \end{cases}\\ \text{ } \\ \text{ } \\ \text{ } \\ \text{Prove } T(n) = n\lg(n) \text{ if } n = 2^k\text{, for } k > 1.$$ I am crawling through the "Introduction to Algorithms" textbook...
  19. T

    Strong Induction Proof with a Floor

    Homework Statement are[/B] an = afloor(n-2) + afloor(2n/3) + n a0 = 1 Prove that for all n ≥ 3, an > 4n Homework EquationsThe Attempt at a Solution Since this is induction, I start out with a base case: Base Case (n = 3): a3 = a1 + a2 + 3 = 3 + 8 + 3 = 14 4(n) = 4(3) = 12 14 > 12 Therefore, the...
  20. Z

    Elastic Potential Energy to Aid Electromagnetic Induction

    Matter can neither be created nor destroyed...but potential energy can be converted into a different kind of energy. Let's say we have a spring with a mass connected to it. This mass is a magnet, and the apparatus is inside a copper coil. It's a horizontal magnet with friction minimized at the...
  21. evinda

    MHB Proving $T(2^k)=2^k \cdot k$ with Induction

    Hello! (Wave) Use induction to show that if $n$ is a power of $2$ then the solution of the recurrence relation$$\left\{\begin{matrix} 2 &, \text{ if } n=2 \\ 2T\left( \frac{n}{2} \right )+n & , \text{ if } n=2^k, \text{ for } k>1 \end{matrix}\right.$$ is $T(n)=n \lg n$.That's what I have...
  22. Colleen G

    Proof by Induction Involving Divisibility

    Homework Statement Let P(n): 7|(34n+1-52n-1. Prove that P(n) is true for every natural number n. Homework Equations *I know that proving by induction requires a proving P(1) true, and then proving P(k+1) true. *If a|b, then b=a*n, for some n∈ℤ The Attempt at a Solution I have proved the "base...
  23. D

    MHB Proof by Induction: Solving 34n-1 Divisible by 80

    As a student, proofs always troubled me, now as a teacher they are still causing me grief. How would you do this one? Prove 34n - 1 is divisible by 80. I understand the process, setting the base and writing as n=k, then n=k+1 etc. But this has stumped me.
  24. dirtybiscuit

    Proof by Induction: Prove Fn-1Fn+1 - Fn = (-1)^n

    Homework Statement The fibonacci numbers are defined by F0 = 0 F1 = 1 and Fn = n-1 + Fn-2 for n >= 2. Use induction the prove the following: Fn-1Fn+1 - Fn = (-1)n The attempt at a solution Let P(n) = Fn-1Fn+1 - Fn2 = (-1)n where n>= 1 Show it holds for first natural number: P(1) = F0 + F2 -...
  25. ElijahRockers

    Fourier series coefficients: proof by induction

    Homework Statement Given f = a0 + sum(ancos(nx) + bnsin(nx)) and f' = a0' + sum(an'cos(nx) + bn'sin(nx)) The sums are over all positive integers up to n. show that a0' = 0, an' = nbn, bn' = -nan Then prove a similar formula for the coefficients of f(k) using induction. Homework EquationsThe...
  26. A

    Electrostatic induction in different size spheres

    if we have a charged sphere with charge Q and radius r, the voltage on it's surface is calculated by gauss's law, what if we approach to it another sphere with radius R , R>r, Earth it, remove earth, basically charging by induction, will the sphere have the same charge, which means less voltage...
  27. M

    Induction cooking & heat distribution

    I've been trying out induction cooking on a small one-burner unit, with a large cast iron pan. I'm getting the expected hot spot in the center of the pan, and a serious temperature falloff toward the sides. No surprise there - the induction element is about 7" (I dismantled the unit to check)...
  28. N

    Proof by Mathematical Induction

    I am confused on a math problem. I am supposed to use mathematical induction to show that (Summation with n on top and i=1 on the bottom) (2i-1) = 1+3+5+...+(2n-1)= n^2
  29. P

    3 Phase Induction Motor Full Load Power Factor

    Hi, If the voltage, full load current and full load output power (kW) of a 3 phase induction motor are known, and the efficiency is assumed to be 90% (is this realistic), am I correct in thinking that the full load power factor can be approximated (due to assumed and not actual efficiency) as...
  30. NATURE.M

    Can Sequence r_n Be Bounded by O(log2(log2 n))?

    Homework Statement We are given the sequence r defined by: r1 = 1, and rn = 1 + rfloor(√n) , n≥2 We need to show by induction that rn is O (log2 (log2 n)). The Attempt at a Solution Definition of big oh: ∃c∈ℝ+, ∃B∈ℕ, ∀n∈ℕ, n≥B => f(n) ≤ cg(n) [/B] So the basic proof format is fairly simple...
  31. A

    Three phase induction motor efficiency

    Hi, Is there a difference in efficiency between a three phase motor and a single phase motor, in terms of power in against power out. If so, could you please explain why one is more efficient than the other. The reason I ask is because I'm researching the development of the induction motor for...
  32. L

    Does an Induction Cooking Pan Need to be Completely Ferrous?

    I ask, "Does a ferrous pan for Induction Cooking need to be Completely Ferrous?" In other words, could just the bottom of the pan be a ferrous material for good inductive heat induction? Could I have a ferrous bottom, and ferrous sides that turn to a non-ferrous material after a couple...
  33. carllacan

    Where is the EMF located in a circuit under induction?

    Suppose we have a closed circuit with a time-varying magnetic flux. An EMF is inducted on the circuit. Where would we put the battery representing the EMF? For example, suppose we have a resistance in the circuit and we want to know the intensity through it. Would we just set the voltage...
  34. evinda

    MHB Prove by induction that the function is injective

    Hi! (Wave) The set $\mathbb{R}$ of real numbers is not countable. Proof: We define the function $F: \{0,1\}^{\omega} \to \mathbb{R}$ with the formula: $$(a_n)_{n \in \omega} \in \{0,1\}^{\omega} \mapsto F((a_n)_{n \in \omega})=\sum_{n=0}^{\infty} \frac{2a_n}{3^{n+1}}$$ Show that $F$ is 1-1...
  35. T

    Induction and flux linkage clarification

    So if you have a single inductor then I assume that there is no flux linkage and that L = flux*N / i = L11 and so that flux is total leakage flux. If you have two inductors then the flux leakage is any flux that isn't flux linkage (the self flux is flux leakage). (flux21 is the flux in 2 from...
  36. 2

    Magnetic fields in transformers and back emf?

    i am having some trouble with understanding the magnetic fields inside transformers and how they relate to back emf. In particular, I am considering the two cases of current passing through the primary coil but not the secondary coil, and then with the secondary coil as part of a complete...
  37. H

    Can Induction Prove 3^n ≥ n2^n for All n ≥ 0?

    Homework Statement The question asks me to prove inductively that 3n ≥ n2n for all n ≥ 0. Homework EquationsThe Attempt at a Solution I believe the base case is when n = 0, in which case this is true. However, I cannot for the life of me prove n = k+1 when n=k is true. I start with: 3^k ≥...
  38. B

    Induction Machines: Torque Speed Characteristics & Benefits

    < Mentor Note -- thread moved to HH from the technical engineering forums, so no HH Template is shown > hello really need some help i would like to know how would the torque speed characteristic for a "constant frequency variable voltage supply" and 'variable frequency variable voltage...
  39. J

    Is an induced emf always produced by a change in flux?

    Faradays law tells us that a change in flux induces an emf. Now consider the phenomenon of motional emf. It is observed across the ends of an open conductor (ie one which is not in a circuit). It is always discussed in connection to faradays law. Where is the change in flux in the case of a...
  40. M

    Inductive transfer via mag. fields, max power transfer?

    Hey guys; I'm looking for info on the maximum power transfer using the magnetic induction method. The current through a coil, then catch the mag field with another coil idea. Mainly looking for high energy transfer methods, maybe some equations that describe it. A friend and i are playing...
  41. Hijaz Aslam

    Charge induced on capacitor plates

    Homework Statement Three conducting capacitor plates are kept horizontally, above each other with a gap 'd'. If a charge ##+1\mu C## is kept on the middle plate what is the charge on the plates? Homework Equations (This question is theoretical) The Attempt at a Solution My textbook gave the...
  42. A

    How does Faraday's law of induction relate to Special Relativity Theory?

    I need to what is induction as it is so confusing. Thanks for the help.
  43. B

    Calculating Induced Voltage in a Flying Airplane through Earth's Magnetic Field

    Homework Statement Assuming you know general specifications of the airplane such as span of the wings and maximum speed, calculate the voltage that is induced when the airplane is flying horizontally through Earth's magnetic field where both the horizontal component and vertical component of...
  44. S

    What is the effect of the capacitor value on an induction generator?

    Can anyone tell me what effect the value of the capacitor in an induction generator has. I am generating 230V, at an unknown speed. I am not particularly concerned about the generated frequency, but I want to run the thing at higher revs, and would like to know whether I need to reduce the...
  45. M

    Proof by Induction: Sum of 2^n - 1 Explained Step by Step

    Homework Statement 1 + 2 + 2^2 + 2^3 + ...+2^(n-1) = 2^n - 1 Homework Equations Sum formula? 3. The Attempt at a Solution I know you are suppose to use the base case and prove its true for k +1, but I'm not sure what to do for similar bases. Can someone please solve it step by step? Thanks!
  46. W

    Squirrel cage induction motors - rotor issues

    Hello all, I am currently involved in a diagnostic project for an induction motor with suspected rotor stator contact. all critical dimensions have been checked but the question has been raised by none electrical observers: Does the location of the rotor bars effect the magnetic center line...
  47. Faradae

    Hysteresis in Induction Cooking

    Hi all, I was researching into Induction Cooking (using induction plates) recently and chanced upon a few websites that mentioned about hysteresis in Induction Cooking. However I don't really get how does hysteresis have a part to play in Induction Cooking. :/
  48. C

    Electromagnetic Induction and a coil of wire

    The magnetic field that is oriented perpendicularly through a 9.2 cm diameter coil of wire drops from 6.4 T to 6.1 T in 0.076 seconds. What is the emf induced in the coil? Trying to answer this. My question is, if they are perpendicular then does that mean that cos90 makes phi = BAcos theta...
  49. evinda

    MHB Proving $T(n)=\Theta(n \log n)$ by Induction

    Hi! (Nerd) I am looking at the following recurrence relation: $T(n)=T \left( \frac{n}{5} \right)+T\left( \frac{4n}{5}\right)+n$. The recurrence tree has this form: So, we see that $T(n)=\sum_{j=0}^{\log_5(4n)} n=n \left( \log_5(4n)+1\right)$ Now, I want to prove by induction that...
  50. S

    Generator / induction formulas help

    Hi , before I ask I want to say that I have done my “ homework” I did a search on this forums and I found a few hits but I couldn’t quite understand them or the numbers turned out wrong after I did my calculations. So I wanted to make a fresh thread , hope you folks will guide me , thanks. So I...
Back
Top