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.

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  1. C

    MHB Prove this proposition 2.1.13 in Induction to Real Analysis by Jiri Lebel

    Dear Everybody, I need some help with seeing if there any logical leaps or any errors in this proves. Corollary 1.2.8 to Proposition 1.2.8 states: if $S\subset\Bbb{R}$ is a non-empty set, bounded from below, then for every $\varepsilon>0$ there exists a $y\in S$ such that $\inf...
  2. C

    MHB Induction Prove: Making Fractions 1/2 to 1

    Suppose you begin with the fraction 1/1. There are 2 rules: a)If you can make a fraction a/b where a/b is in its lowest terms, then you can also make b/2a. b)If you can make a/b and c/d where they are both in lowest terms, you can also make (a+c)/(b+d). Prove that you can make all fractions...
  3. V

    Doubt in electromagnetic induction

    Homework Statement suppose a coil is placed in a changing magnetic field and the circuit is not closed will the current induce in the coil Homework EquationsThe Attempt at a Solution What I thought was that the current flows only in a closed circuit because it needs a potential difference...
  4. T

    How Does the Feynman Paradox Apply to Solenoid Flux Calculations?

    I am currently going through a friend's Mechanics II notes and homework before I take the course at a different university next semester. I have a few problems that I am having trouble understanding and am posting here for help. 1. Homework Statement 1. A solenoid is on, Flux is Φ. 2. The...
  5. FlimFlam

    Grass Death Under Power Lines: EMF's a Factor?

    So I was messing with the induction from power lines and fluorescent bulbs, and I noticed all the plants under the power lines were dead but plants around it were fine. Do EMF's have something to do with this?
  6. J

    Movement by electrical induction

    So, when you bring a charged object near for example a soda can, I get that the soda can gains a positive side and a negative side, but the can also moves towards the charged object, why is that? Is it because if you put a negative charged object near the soda can the side of the can nearest to...
  7. LonelyElectron

    Electromagnetic induction questions

    Problem: Look at the following diagram: · Copy the diagram and then label the poles for the induced magnetic field and the direction of the induced current · Explain what would happen to the magnetic field and the current if the direction of motion of the magnet is reversed ·...
  8. Monoxdifly

    MHB [ASK] Induction Determine the value of n if 1 + 3 + 6 + .... + n(n - 1)/2 = 364

    Determine the value of n if 1 + 3 + 6 + ... + \frac{1}{2}n(n - 1) = 364 What I did: I know that 1, 3, and 6 are the result of arithmetic series with the starting value 1 and the difference 2, thus that sum can be written as S1 + S2 + S3 + ... + Sn = 364. However, by assuming that Sn =...
  9. D

    I Understanding the Induction Axiom: Notation & Equivalence

    So , what I was wondering about was a slight difference in notation, for which I am not certain if correct (mine, in particular.). The induction axiom says: If M is a subset of ℕ, and if holds that: a)1∈M b)(∨n∈ℕ)(n∈M→s(n)∈M) then M=ℕ. Now my question is: why do we write (∨n∈ℕ)(n∈M→s(n)∈M)...
  10. Monoxdifly

    MHB Proving Divisibility by 6 Using Mathematical Induction

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  11. R

    Induction Proof for A^n = 1 2^nProve your formula by mathematical induction.

    <Moderator's note: Moved from a technical forum and thus no template.> A^n = 1 2^n 0 1 Prove your formula by mathematical induction. I began by taking A to successive powers but not sure of what my formula should be. A^0 = 1 0 , A^1 = 1 2 , A^2 = 1 4 , A^3 = 1 6 ...
  12. rishi kesh

    Engineering Resultant magnetic flux in a 3-phase induction motor

    Homework Statement I have some attachements below from my textbook. According to it the value of resultant flux is constant in magnitude and 1.5 times the maximum value of flux. But i have a doubt regarding how it is proved. The magnetic flux waveforms are 120° apart from each other and at...
  13. D

    Calculating magnetic induction in a triangle

    Homework Statement Three very long parallel conductors situated in the air make a direct-symmetrical 3-Phase system. The conductors pass through the A B C points of the triangle of side ##a##. The currents in the conductor form a direct-symmetrical 3-Phase system. Effective values of currents...
  14. T

    Confusion about induction in conductors

    Homework Statement Homework EquationsThe Attempt at a Solution a) charge induced is ##\frac{-q_i}{4 \pi r_i^2} = \sigma_i## ##\frac{q_a + q_b}{4 \pi R^2} = \sigma_R## b) field outside is ##\frac{k (q_a + q_b)}{r^2}## c)i naively applied gauss law to get field within each cavity as...
  15. J

    Engineering Single Phase Induction Motor -- 2 Questions

    Homework Statement Question 2 Homework Equations Flux time lag depends on L/R ratio. More resistance, less time lag. Single Phase IM---Double Revolving field theory The Attempt at a Solution For 1st question, I think answer should be A. BB circuit has more resistance so BB Flux will have...
  16. S

    Oscillation related to electromagnetic induction

    Homework Statement A copper ring is suspended by a long, light rod pivoted at X so that it may swing as pendulum, as shown in the diagram below. An electromagnet is mounted so that the ring passes over it as it swings. The ring is set into oscillation with switch K open. What happens to the...
  17. Pushoam

    Angular momentum due to electromagnetic induction

    Homework Statement Homework EquationsThe Attempt at a Solution ## \frac { - d \phi }{dt} = V ## V denotes emf. The current is in ## \hat \phi ## direction. Magnetic force is along ## ~\hat s ## direction. Where ## ~\hat s ## is the radially outward direction in cylindrical...
  18. Aya Elsayed

    The circuit of the induction coil

    When we turn the circuit off . There will be an induced emf through the coil due to the changing magnetic field of the AC . Therefore V=-L . DI/Dt So when the current is zero then the emf has a maximum value . And what about the emf of the source itself ? It's equal to the induced emf? If It...
  19. VSayantan

    Speed of a Magnet Falling through a Conducting Wire

    Homework Statement A small but very powerful bar magnet falls from rest under gravity through the center of a horizontal ring of conducting wire, as shown in the figure below (on the left). The speed-versus-time graph, in arbitrary units, of the magnet will correspond most closely to which of...
  20. F

    B What's the relationship between drop height and induced emf?

    In my experiment, I intended to find out how the change in the bar magnet drop height from solenoid affected the emf induced in the solenoid, however, I am unable to come up with an equation that shows a relationship between the two variables. I have thought of Biot-Savart law, but I do not...
  21. J

    MHB Proof by Induction - in Sequences.

    Dear ALL, My last Question of the Day? Let b1 and b2 be a sequence of numbers defined by: b_{n}=b_{n-1}+2b_{n-2} where $b_1=1,\,b_2=5$ and $n\ge3$ a) Write out the 1st 10 terms. b) Using strong Induction, show that: b_n=2^n+(-1)^n Many Thanks John C.
  22. P

    How does the differential form of Faraday's law relate to induced current?

    Homework Statement I'm in a class where we have to essentially learn E&M ourselves and I'm challenged by Maxwells equations. I'm studying out of Purcell's E&M. The differential form of Faradays law for Maxwells equations is curl E = -∂B/∂t Im having trouble interpreting what to make of this...
  23. S

    Prove usingStructural Induction

    Homework Statement [/B] Let the set S be defined recursively as follows: Basis Step: (0, 0, 2) ∈ S Recursive Step: If (a, b, c) ∈ S, then (a + 1, b + 1, c) ∈ S and (a+1, b, c+1) ∈ S Use structural induction to prove that a + b + c is even when (a, b, c) ∈ S The Attempt at a Solution...
  24. T

    Is it a magnetic or an electric field that causes induction

    Is it a magnetic or an electric field that causes induction in the antenna wiki says that: The electric field (E, green arrows) of the incoming wave pushes the electrons in the rods back and forth, charging the ends alternately positive (+) and negative (−). enter image description here...
  25. T

    Why does electromagnetic induction occur

    Greeting I know the field of electromagnetism is not yet fully clarified but I wonder if there is an idea,theory Why does electromagnetic induction occur only when the conductor cuts the magnetic field lines.
  26. M

    MHB Double Induction: Proving Addition is Commutative

    Hey! :o I am looking for an example of an application of the double induction. For that I tried to show that the addition is commutative: For $m,n\in \mathbb{N}$ it holds that $m+n=n+m$. I have done the following: For each $n\in \mathbb{N}$ let $E_n$ be the proposition: $\forall m\in...
  27. R

    Adding a 2nd Secondary coil & effects

    Hi, We have an induction heater, like a stove top one, not industrial. I want to understand what happens when an additional load is added that it isn't intended to receive. A plate goes on the induction warmer, it gets heated up, then removed and another plate takes it place. What happens if the...
  28. F

    Induction motor - calculating rotor flux angle

    Hello. I'm trying to write my own soft for Field Oriented Control of induction motor. I have finished hardware - 3ph 3kW inverter, controlled by STM32F4, and V/F open loop algorithm to control ACIM - it works. Now I'm trying to implement FOC algorithm. As we know we need to do Clark&Park...
  29. K

    Inducing electricity w/ moving piece of metal, static magnet

    Hi! I'm curious to know, whether it is possible to produce electricity by moving a piece of metal over a magnet (i.e a immobile magnet and a moving piece of metal close to the magnet), and how this would be done in practice.
  30. S

    Direction of the induced current

    All sites and books give different answers for this question:( Homework Statement predict the polarity of the capacitor in the diagram below Homework Equations right hand thumb rule AB is capacitor The Attempt at a Solution The side facing south pole becomes south and north facing side...
  31. B

    Induction via Infinite sheet of Current

    My thoughts are concerning an infinite sheet carrying current in a direction. I wish to induce a unidirectional electric field outside of the sheet via induction. My ideas were that if the current was changed linearly in a sawtooth fashion, then I would achieve the induced electric field in...
  32. M

    Questions about induction heating

    Hi there, I'm a material chemist, currently working on ceramics. I know the very basics of physics, but I have many doubts about electromagnetism. How does induction heating exactly work? I understand that you use a hollow coil of copper with a cooling fluid circulating inside, through which...
  33. T

    Can the Sequence \( a_n \) Satisfy the Inequality \( a_n \leq 20n \)?

    Homework Statement a0 = 0, and for n > 0, $$a_n = a_{\frac {n} {5}} + a_{\frac {3n} {5}} + n $$ For the above equation, besides an, the subscripts are floored Prove that an ≤ 20n Homework Equations See above. The Attempt at a Solution I know how to do the question, my problem is starting...
  34. T

    Induction and the Fibonacci Sequence

    Homework Statement Define the Fibonacci Sequence as follows: f1 = f2 = 1, and for n≥3, $$f_n = f_{n-1} + f_{n-2}, $$ Prove that $$\sum_{i=1}^n f^{2}{}_{i} = f_{n+1} * f_{n} $$ Homework Equations See above. The Attempt at a Solution Due to two variables being present in both the Sequence...
  35. Noaha

    Rotor and stator flux in 3 phase Induction motor

    The rotor and stator flux speed are stationary with respect to each other in 3 phase induction motor. What will happen if there is relative speed between them? How these two fluxes interact with each other to develop a torque?
  36. A

    Magnetic/electromagnetic induction

    Suppose there is a coil around a C-shaped iron core with a dipole magnet in the gap as shown in the dynamo below. As the magnet is spinning, in order to oppose the force, a changing pole is produced in the iron. (Eg. As N-pole approaches the iron core, a north pole is induced to oppose the...
  37. M

    Prove ##5^n+9<6^n## for ##n\epsilon N|n\ge2## by induction

    Homework Statement Prove ##5^n+9<6^n## for ##n\epsilon \mathbb{N}|n\ge2## by induction. Homework Equations None The Attempt at a Solution The base case which is when ##n=2##: ##5^2+9<6^2## ##34<36## Thus, the base case is true. Now for the induction step. Induction hypothesis: Assume...
  38. parshyaa

    Question about mathematical induction

    In mathematical induction we prove statements using a proper technique just like in a following example: ##P(n)=n(n+1)## is an even number for every ##n∈N## we will check wether ##P(1)## is True/false, its true because ##P(1)=2## Now We will assume that ##P(k)## is true and using this we will...
  39. B

    WEG IE4 induction motor design

    Hello, is there anyone who knows how WEG managed to get IE4 efficiency rating for their motors while keeping Aluminum for cage on rotor? http://ecatalog.weg.net/tec_cat/tech_motor_sel_web.asp# Best regards
  40. Pushoam

    Faraday's law of electromagnetic induction -- Motional EMF

    There is magnetic field in B = B ##\hat z## in a region from x=0 to x=l. There is a metal rectangular wire loop with length l and width w in x- y plane with coordinates of four corners as (0,0),(0,w),(l,0),(l,w). This loop is moved with velocity v=v##\hat x##. Now according to Faraday's law...
  41. T

    VSD for induction motor maximum current

    Hi, I'm trying to ascertain when you're driving an asynchronous motor with VVVF or whatever else a VSD uses, when do you draw the most current? For instance, say you're talking about a train/tram: I would have thought intuitively that since power = torque / angular speed, that starting up, to...
  42. B

    Solving Symmetric Group Induction Proof: Hints for Double Induction

    Homework Statement Consider the symmetric group ##S_n##. I am trying to establish that ##(i,i+1)=(1,2,...,n)(i-1,i)(1,2,...,n)^{-1}## Homework EquationsThe Attempt at a Solution I am trying to decide whether I need double induction or not. I have done several calculations to see whether I can...
  43. Vikas Kasireddy

    Single Phase Induction motor winding phase difference

    Q1: So I'm trying to extract the electrical properties of a single phase induction motor. First off, how am I supposed to measure the phase difference between the start and run windings? I have L and R of start and run windings too! Q2: I have a torque vs slip curve up until 130% of rated load...
  44. B

    Jackson problem 5.4: Magnetic induction in a current-free region

    Homework Statement Homework EquationsThe Attempt at a Solution I think that is taylor series expansion, How that can be expanded right that? even b0, b1 ... is not a differential term. The solution just use b(z) = Bz(0,z)
  45. B

    Simple calculation about Magnetic induction

    Homework Statement Homework EquationsThe Attempt at a Solution Why does the second term become zero??
  46. D

    Electromagnetic Induction repulsion

    Homework Statement http://imgur.com/a/7CUcU What happens when you close W? What happens when you open W again at Q? If there is a ring of metal at Q, on the coil. Homework Equations There is no relevant equations in this case. The Attempt at a Solution North will be on top and south on the...
  47. Ruturaj Vaidya

    Induction Heating: Calculating Magnetic Field from Temperature

    Hi everyone, I'm conducting an experiment where I place a ferrous material rod inside a solenoid to measure it's temperature due to induction heating, as shown in the video below: The aim of my experiment is to calculate the magnetic field from the temperature of the ferrous material rod. Is...
  48. F

    Prove Induction: u_n < 4 for All n ≥ 1

    Homework Statement The sequence of positive numbers ##u_1,u_2,u_3...## is such that ##u_1<4## and ##u_{n+1}= \frac{5u_n+4}{u_n+2} ## i. By considering ##4-u_{n+1} ##, prove by induction that ##u_1<4## for ##n\geq 1## Mod note: The above is incorrect. In a later post the OP revised this to The...
  49. P

    Electromagnetic induction in current loops

    would be really grateful if someone could help throw some light on this - its probably really simple but I am typing myself up in loops! I have a piece of wire which is placed in an external AC magnetic field which is varying sinusoidally. The wire is folded to make two rings of radii r1 and r2...
  50. Eclair_de_XII

    Is my short induction proof correct?

    Homework Statement "Prove: ##∀n∈ℕ##, ##3^n>n^2## Homework EquationsThe Attempt at a Solution (1) We will prove that ##3^n>n^2## at ##n=1## ##3=3^1>1=1^2## (2) Now assume that ##3^k>k^2## for some ##k>1## (3) We will prove that ##3^{k+1}>(k+1)^2## or ##3⋅3^k>k^2+2k+1## Note that...
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