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. J

    Proving the Number of Leaves is One More Than Internal Nodes in Binary Trees

    Homework Statement In a binary tree all nodes are either internal or they are leaves. In our definition, internal nodes always have two children and leaves have zero children. Prove that for such trees, the number of leaves is always one more than the number of internal nodes. Homework...
  2. J

    Induction Proof for 2^n x 2^n Matrix Using L Transformation

    Homework Statement Attached is the problem Homework EquationsThe Attempt at a Solution The trick to solve this problem is that when we assume that it is true for a 2^n x 2^n matrix and then we expand this matrix with 1's to a 2^n+1 x 2^n+1, we can divide the resulting matrix into 4 submatrices...
  3. R

    Induction Motor Builds & Experiments: What PSU to Use?

    So if you wanted to do some induction motor builds and experiments, what PSU would you use. I guess I am looking for a bench top AC multi output system that has ability to supply out of phase 2nds and maybe 3rd??
  4. H

    Explanation please -- public bus induction power -- into basic physics?

    Sorry for the title, and sorry if there is already a thread on it. This induction charger for a public transport bus. Can anyone break it down into basic formulas of high school physics. The air gap is frikkin huge. Earlier prototypes apparently had a secondary coil on the bus that was lowered...
  5. Z

    How Is Magnetic Induction Calculated in a Triangle of Parallel Wires?

    Homework Statement Three parallel wires are put in the angles of an equilateral triangle with sides 14 cm. The values of the currents are 10A,10A, 20A and the first and third are in different directions. Find the magnetic induction in the center of the triangle Homework Equations B=I*μ0/2*pi*d...
  6. Z

    Find the induction of the magnetic field at the point O

    Homework Statement In a wire passes the current I=21A. A part of the wire is bent to form three consecutive sides of a square with sides 40 cm. Find the induction of the magnetic field in the point O which is the center of the square. Homework Equations B=I*μ0*/2*pi*d The Attempt at a Solution...
  7. L

    Equivalent circuit of an Induction Motor

    Hi, I have attached two circuits of an induction motor, one for steady state(1st circuit) and a dynamic model(2nd circuit). My question is: Why don't we include the induced emf in the steady state circuit? Looking at the dynamic circuit we include it, but not in the steady state circuit. Is the...
  8. erashish14

    Electromagnetic Induction Help Requested

    Hi Experts, I'm doing an experiment of Electromagnetic Induction (producing current by changing magnetic field i.e. by circulating magnetic wheel of N52 Neodymium Magnets around copper coil), but what i found when the speed of wheel is HIGH, the Output Current is LOW (in range of .02...
  9. Battlemage!

    I Can you use proof by contradiction in the midst of induction

    In the process of doing a proof by induction, can you use a contradiction to show that if P(k) holds then P(k+1) must hold? What I mean is, after establishing that P(0) holds, can I assume that P(k) holds and that P(k+1) does not, and show that a contradiction arises, and thus conclude that if...
  10. srinaath

    Why slip is more in low HP induction motor?

    am doing a comparative study of induction motors (both single and three phase) based on cost, size, efficiency,etc... while analyzing 1/4Hp and 5 HP motors, i read that 1/4HP motor has high slip than 5HP motor because 1/4HP motor has greater rotor winding resistance than 5HP...so efficiency is...
  11. Luek

    Energy stored in a LC circuit

    Homework Statement Energy in the circuit remains constant. When the current is flowing, the energy stored is all stored in the inductor. When the current stops flowing, it is because all the energy is stored in the capacitor. Find the fraction of energy stored in the inductor when half of the...
  12. srinaath

    Why single phase induction motors are used for fractional HP applications

    why single phase induction motor is commonly used for low HP rather than integral HP like 1,2,3HP... i understand that single phase gives low power...but say you have designed such that your motor delivers 10HP for rated voltage and current...why such single phase machines are not common?can...
  13. srinaath

    Why efficiency of induction motor is low for low HP?

    why there is a decrease in efficiency of induction motor when power rating is low? example 1 HP motor efficiency is say 85% then 1/4HP motor efficiency is less than 85%? please clear me this doubt?
  14. E

    Solve Homework 2: Electromotive Force in a Shrinking Loop

    Homework Statement A little stuck on the second question it basically boils down to not quite understanding the question. Was hoping someone here could push me in the right direction. The Question is two part, already completed the first half but wouldn't mind if someone looked over it while...
  15. B

    Induction motor control by DTC method

    I´m simulating an induction motor control using DTC (direct torque control). I want to tune my PI controller to achieve a fast start up, but I don't have a transfer function to make an analysis. How can I set this controller?
  16. srinaath

    Cost of three phase and single phase induction motor

    hi guys... which will be costlier-single phase induction machine or three phase induction machine for same power rating, same rpm speed, same input voltage and current.?
  17. UncertaintyAjay

    Current Growth due to Magnetic Induction

    What factors affect the rate of growth of current when an emf is induced in a coil?
  18. M

    Magnetic Induction in Two Different Media

    From Faraday's law of induction we know if we have a coil with ac current it will create ac current to other coil placed within magnetic field. My question is if one coil is in water and another coil is in air how I can calculate the induced emf?
  19. Z

    In Induction Heating, what does the heat output depend on?

    Wasn't sure whether to post this here or in EE forum. I've been learning about induction heating for the past few hours and scoured every video on youtube and every web article I could come across and I still lack an understanding of how to determine the heat output of an induction heater. I...
  20. srinaath

    Driving three phase induction motor with single phase supply

    hi guys, am working on variable speed drive system. My aim is to make a low cost drive system for three phase induction motor. Current system employs three phase AC-DC and DC-AC converters, which has 12 switches in total. I have one idea...i found three phase inverter topology which has 4...
  21. Kingyou123

    Induction Proof Help: Understanding Equations for Homework | Prof. Note's Tips

    Homework Statement Homework Equations Prof. Note's. The Attempt at a Solution I'm on the 3 line where my Prof. combines both equations, I'm confused on what my equation should look. Her's was (n+1)(n+1)+1)/2
  22. E

    Induction heating by AC magnetic field-magnetic hyperthermia

    Hi, I want to heat magnetic nanoparticles in an AC magnetic field. I need a uniform and a magnetic field about 15- 25 mT ) . I have thought about some designs, can you tell me which one is the most suitable design for creating a uniform alternating magnetic field? I need this system for...
  23. T

    Clean way to run 3 phase induction motor

    Say you wanted to run an induction motor from a battery without putting dirty harmonics into it. Would a really clean way to do it, to be by using a higher phase number inverter, then transformer, then into the motor? For instance, a 6 phase inverter, into a 6 phase to 3 phase transformer, then...
  24. J

    Equivalent Circuit for a CSCR Single Phase Induction Motor

    I have created a circuit diagram for a single phase CSCR induction motor, but only for the purpose of calculating start-up current (i.e. motor speed is at, or very near, zero RPM). I have created this diagram because: 1) I do not have the tools & knowledge required to generate the Steinmetz...
  25. B

    Solving a Recurrence Relation Using Induction: Step-by-Step Guide

    Below I have uploaded the page I am having trouble with. Here it says that it is using induction on n but I don't understand how it uses the formula for when n=j to derive the formula for when n=j+1
  26. E

    Electromagnetic induction graph

    Homework Statement A small bar magnet is being slowly inserted with constant velocity inside a solenoid as shown in figure. Which graph best represents the relationship between emf induced with time Homework Equations ε = -dφ/dt φ = B.A The Attempt at a Solution Since the magnet is brought...
  27. G

    What sort of EM field is produced in an induction heater?

    What sort of EM field is produced in an induction heater? And how to measure it? That's it, basically. Assuming an induction heater is running on 200khz, and consuming about 3000 watts, how would I go about measuring the magnetic field produced in its coil? Now, I'm rather confused about all...
  28. ranju

    Engineering What is the role of rotor bars in single phase induction motors?

    Homework Statement I studied that as the single phase motor speeds up the backward induced field currents are larger than the forward field currents. So , the corresponding opposing rotor mmf causes backward field to be greatly reduced whereas the smaller forward currents leads to enhancement...
  29. T

    Proving an equality using induction proof not working

    Homework Statement I work out the problem completely and it does not equal out. Having problems with two variable induction proofs (n and k) in this problem. Below is as far as I got, jpeg below Homework EquationsThe Attempt at a Solution
  30. A

    Electromagnetic induction by a rotating sphere

    Homework Statement http://physics.columbia.edu/files/physics/content/Quals2010Sec2.pdf Problem 1 Consider a rigid, ideally conducting sphere of radius R, with total charge zero. The sphere rotates with angular velocity Ω, ΩR<<c. Suppose a dipole magnetic field threads the sphere. the dipole is...
  31. M

    Is it possible to detect current ramp-up to DC with a coil?

    Hi, I am thinking of ways to detect when an electrical device is turned on and off without actually touching or modifying the existing circuitry. More specifically, I have a hand held device powered by 2 x 9V batteries and with 2 electrodes connected through leads and plugs to sockets on the...
  32. Fredrik

    Theorems that can't be proved by induction

    I know that I've seen an example of a statement of the form ##\forall n~P(n)## (where the scope of the "for all" is the set of positive integers) that can be proved, but can't be proved by induction. I thought I had seen it in one of Roger Penrose's books, but I have looked for it and wasn't...
  33. S

    Energy loss when boiling in an water experiment -- help please

    Homework Statement Hi, as a part of my lab report I have to conduct this experiment : Fill a pot with tap water and boil it, determine then how much of the energy that the kitchen surface produced, actually went to the water itself. Consider the water having an initial temperature of 10 °C. In...
  34. ELiT.Maxwell

    Point charge inside a electrically neutral cavity in conduct

    bear with me, i know that this question has been asked many time , but i would like a definite answer, now, starting off the external charge density on the outer surface of sphere WILL be uniform by unique solution of Laplace equation and letting the sphere be huge, so, electric field due to...
  35. S

    Question about Faraday's law of induction

    I was studying the Maxwell equation for Faraday's law: ∇×E = -(∂B/∂t) I then did some math and noticed that the electric field is a conservative vector field, because ∇×E= <0,0,0> Since this is the case, based on the above Maxwell equation this would set the time derivative of the magnetic...
  36. cnh1995

    Rotor EMF in squirrel cage induction motor

    In a 3 phase induction motor,when stator rmf rotates, it cuts the rotor bars and induces an emf in them. Is this emf sinusoidal? I mean shouldn't it be a square wave since the rotor "bars" are being cut by the flux, like in case of concentrated stator windings of a synchronous generator? If the...
  37. L

    Prove by induction: 2^n >= 11n + 17

    Homework Statement Prove by mathematical induction: 2^n >= 11n + 17, for n >= 7, and n is an integer. Homework EquationsThe Attempt at a Solution This is my attempt - I want to see if I'm doing this correctly. 2^n >= 11n + 17, n >= 7 basis 2^7 >= 77 + 17 128 >= 94. True. Induction...
  38. J

    Engineering Magnetising circuit parameters in a 3ph induction motor

    Homework Statement Homework EquationsThe Attempt at a Solution Stator Line voltage 220 164 112 88 42 Stator Line Current 6.8 5.4 3.9 3.8 3.7 Stator power 470 360 278 244 232 By plotting a suitable graph from these results, determine the total of windage and friction...
  39. srinaath

    Supply 460v single phase instead of 230v to induction machin

    I am planning to run 230V single phase machine with 460V single phase supply. what will be the effects in RPM and torque? (( this is how i thought: as we increase voltage , current will decrease since power is same...decrease in current decreases the torque...to maintain same power, decrease in...
  40. J

    Electrical reason for big air gap of induction motor

    in induction motor there is air gap between stator and rotor. the air gap needs to be as small as possible. but some air gap is needed for mechancial reasons. i understand that a big air gap would reduce power factor as you'd need more magnetizing current. but are there any advantages of big...
  41. cnh1995

    Why Does Torque Increase as Slip Decreases in a Single Phase Induction Motor?

    Single phase motor is not self starting, so it is started using some arrangement and the starting arrangement is disconnected(not always) once the motor is set in proper motion(about 70% of the maximum speed). But I didn't understand this graph. From the starting moment, motor accelerates and...
  42. T

    What Happens to Flux in an Induction Motor When Short Circuited?

    Just a curiosity about what happens to the flux, when you short circuit an induction machine. Say there is an initial flux and you're rotating the shaft of an IM and then you short circuit the stator, so no doubt you'll get large currents in the stator, but those large currents, will they cause...
  43. T

    Induction machine slip control

    Hi, Say you wanted to have a generator or motor stay at near 90degrees for maximum torque while generating or motoring, over various loads or over various supply frequencies. I assume there is some established method of controlling this (varying the rotor resistance with feedback perhaps?) does...
  44. ognik

    MHB Induction Proof: Sum of Series $ \frac{1}{(2n-1)(2n+1)} = \frac{1}{2}$

    Q. Show by induction that $ \sum_{1}^{\infty} \frac{1}{(2n-1)(2n+1)} = \frac{1}{2} $ So, start with base case n=1, $ S_1 = \frac{1}{(2-1)(2+1)} = \frac{1}{3}$? Maybe it's bedtime ...
  45. cnh1995

    Cascade connection of induction motors

    In cumulative cascade connection of two induction motors, why does the net speed reduce? Also, how are the torques of the two motors in the same direction?
  46. R

    Electromagnetic Induction concept understanding

    Homework Statement Using a galvanometer and a coil, we thrust a magnet with a defined north and south pole into the coil and the galvanometer measured the induced current created. Now, depending on how the coil flow(front to back or vice versa) the galvanometer measured an induced current...
  47. cnh1995

    Understanding the Formation of 4 Poles in an Induction Motor

    I've studied a 2-pole IM with its winding diagram and understood how the two poles are formed. But I'm struggling to understand the same thing about a 4-pole motor. How does this winding arrangement result into 4 poles?
  48. V

    Direction of current in a magnetic field when loop is square

    Homework Statement A flexible wire loop has a radius of r = 0.178 m and it is inside a uniform magnetic field of B = 0.332 T. The loop is grasped at points P and Q and stretched until its area is zero. It takes 0.146 s to close the loop. Homework Equations EMF = ∆Phi/∆t Phi = magnetic flux...
  49. H

    What rules governs the mutual induction

    Homework Statement Homework Equations With what equations i can express the fact that when dc is aplied to transformer there is mutual inductance only when the voltage is turned on or off and there is no positive feed back represented as an infinite inducyion between the 2 coils? The Attempt at...
  50. 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 >=...
Back
Top