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. I

    Maxwell's Law of Induction - How Does it Work?

    So, I was studying Maxwell's equations and I don't really understand the last one - Ampere's Law (with Maxwell's extra term added in). The bit I'm not able to understand is the term Maxwell added. How exactly does a changing electric field through a closed loop induce a magnetic field along that...
  2. C

    MHB Recursive definition and induction

    Hey. The series $a_n$ is defined by a recursive formula $a_n = a_{n-1} + a_{n-3}$ and its base case is $a_1 = 1 \ a_2 = 2 \ a_3 = 3$. Prove that every natural number can be written as a sum (of one or more) of different elements of the series $a_n$. Now, I know that is correct intuitively but...
  3. Entertainment Unit

    Help with an arithmetic proof by induction

    Homework Statement Show that if ##0 \leq a < b##, then $$\frac {b^{n + 1} - a^{n + 1}} {b - a} < (n + 1)b^n$$ Homework Equations None that I'm aware of. The Attempt at a Solution Proof (Induction) 1. Basis Case: Suppose ##n = 1##. It follows that: $$\frac {b^{1 + 1} - a^{1 + 1}} {b - a} < (1 +...
  4. Celso

    Induced EMF — generator coil area calculation

    Homework Statement What is the necessary area for a generator that produces an emf of ##\mathcal{E} = 150V## when it spins at a ratio of 60 revolutions per second, in a magnetic field of ##B = 0.5 T##? Homework Equations ##\oint_{c} E \cdot dl = \mathcal{E} = -\frac{d}{dt}\iint_{s} B \cdot dS...
  5. D

    Proof by Induction: Proving n=i=0n∑ ai2i

    <Moderator's note: Moved from a technical forum and thus no template.> for every natural n there exists natural k. and numbers={a0,a1,a2,...ak}∈{0,1}. so that n=i=0n∑ ai2i I will assume n=k, i know that if n is even then a0 =0. so if i assume it is true for n that is Even: n+1=i=0n+1∑ ai2i...
  6. T

    MHB Proof by induction - Really confused

    Hi again. I have one other problem I'm puzzled about. (a) A sorting algorithm takes one operation to sort an array with one item in it. Increasing the number of items in the array from n to n + 1 requires at most an additional 2n + 1 operations. Prove by induction that the number of operations...
  7. S

    Induction Motor Auto-Transformer Starter - Power and Torque

    Homework Statement Describe an auto-transformer starter. From the expression for power input per phase (∝ torque) show that for a turns ratio (n) of √3:1 , for the auto-transformer: Homework Equations Pin (phase) = 3 V I cos∅ I = V/Z cos∅ = R/Z T∝I∧2 T∝V∧2 For auto transformer with...
  8. FritoTaco

    How to Use Mathematical Induction to Prove Equations?

    Homework Statement Prove that 1\cdot1!+2\cdot2!+...+n\cdot n! = (n+1)!-1 whenever n is a positive integer. Homework EquationsThe Attempt at a Solution I'm having trouble simplifying towards the end of the proof. Proof: Let P(n) be the statement 1\cdot1!+2\cdot2!+...+n\cdot n! = (n+1)!-1...
  9. L

    Electromagnetic Induction and Faraday's Law w/ Right Hand Rule

    Homework Statement Homework Equations Right Hand Rule The Attempt at a Solution I am not understanding why the force is left. I can only figure out that the current in the solenoid is moving clockwise because of the right hand rule. From there, I see that the induced current might be...
  10. S

    Prove by mathematical induction Σ(1/[(2k-1)(2k+1)]=n/(2n+1)

    I'm in need of assistance for the following attachement.
  11. S

    MHB Solve Prove by Induction: 3^2n+1 + 2^n-1 Divisible by 7

    Can someone with understanding of proof by induction help with this problem? Prove by induction that 3 raised to 2n+1 + 2 raised to n-1 is divisible by 7 for all numbers greater than/or equal to 1. How do you do the inductive step?
  12. M

    Proof by induction (summation)

    Homework Statement Prove by induction that ##\sum\limits_{k=1}^{2n} \frac{1}{k(k+1)} = \frac{2n}{2n+1}## 2. The attempt at a solution First I showed that it is true for ##n=1## and ##n=2##. Then, assuming it is true for all ##n##, I attempt to show that it is true for ##n+1##...
  13. M

    MHB Show by induction that the determinant is equal to n

    Hey! :o For $n\in \mathbb{N}$ let $A_n$ be the real $n\times n$-matrix with the elements \begin{equation*}a_{ij}=\begin{cases}i , &\text{ if } i=j-1 \\ 1, & \text{ if } i=j \\ -j, & \text{ if } i=j+1 \\ 0 , & \text{ otherwise } \end{cases}\end{equation*} For $n=1, 2, 3$ we get the matrices ...
  14. E

    Proof by Induction of shortest suffix of concatenated string

    Homework Statement Wherein α, β are strings, λ = ∅ = empty string, βr is the shortest suffix of the string β, βl is the longest prefix of the string β, and T* is the set of all strings in the Alphabet T, |α| denotes the length of a string α, and the operator ⋅ (dot) denotes concatenation of...
  15. C

    MHB Math induction with sigma notation

    Prove by math induction that n sigma 3i + 1 = n/2 (3n + 5) i = n
  16. E

    Proof by Induction of String exponentiation? (Algorithms)

    Homework Statement Wherein α is a string, λ = ∅ = the empty string, and T* is the set of all strings in the Alphabet T. Homework Equations (exp-Recursive-Clause 1) : α0 = λ (exp-Recursive-Clause 2) : αn+1 = (αn) ⋅ α The Attempt at a Solution [/B] This one is proving difficult for me. I...
  17. U

    Proof via mathematical induction

    Homework Statement Use mathematical induction to prove that (8n − 7n − 1) is divisible by 49 for any n ∈ N. Correction by mentor for better readability: ##49\,|\,(8^n-7n-1)## The Attempt at a Solution We can see that the base case is satisfied here: n = 1, 8^1-7*1-1 = 0 and 49 | 0 is true...
  18. mheruian

    Winding Direction Consideration (Proper Winding)

    Hello, I'm boy who suddenly been strucked by curiosity on how hand crank generator works and i watch and read some of how to create one, specially this one on youtube: and i want to create one but i don't have a 3D printer, but i saw some alternative but as i do/design my own coil based on...
  19. K

    Electronics Temperature Sensing to Switch on/off an Induction Heater

    Hello, I'm having trouble thinking of ways that I can measure the temperature of an object that I am heating using an Induction Heater. A little background: The coil and workpiece will be submerged in liquid, and due to most thermometers being metal it won't work with the induction heater. I'm...
  20. U

    Calculation of Change in Magnetic Flux Linkage Across a Wire

    Homework Statement A straight wire of length 0.20m moves at a steady speed of 3.0m/s at right angles to a magnetic filed of flux density 0.10T. Use Faraday's law to determine the e.m.f. induced across the ends of a wire. Homework Equations E= Nd Φ/dt but N=1 so E= dΦ/dt The Attempt at a...
  21. R

    Solar PV:What happens to the lost energy?inductive recovery?

    Hi there, I am new in here, thanks for any reply. I took same title of a previous closed 3D by pranj5. As understood main part of the energy absorbed by PV panels goes in reflection and heat. The heat origins mainly in electrons that are excited but not enough to jump or they jump but they...
  22. Mr Davis 97

    Proving Induction for All Integers

    Homework Statement Let ##\phi : G \to H## be a homomorphism. Prove that ##\phi (x^n) = \phi (x)^n## for all ##n \in \mathbb{Z}## Homework EquationsThe Attempt at a Solution First, we note that ##\phi (x^0) = \phi(x)^0##. This is because ##1_G \cdot 1_G = 1_G \implies \phi (1_G 1_G) = \phi...
  23. Mathmellow

    MHB Understanding induction proof of pigeonhole principle

    I am struggling to understand the induction proof of the pigeonhole principle in my textbook. The theorem and the proof, from Biggs Discrete Mathematics, is pasted below, and I will explain further (see bold text) what I am having trouble with. Theorem. Let m be a natural number. Then the...
  24. T

    Simple Question about the direction of magnetic induction

    Homework Statement There is a magnetic field B=0.02 Tesla and its direction is toward the plane (perpendicular to plane, shown by X X X X X - I'm not sure how to say this direction in English,sorry). we have a copper wire (uniform bar) perpendicular to magnetic field from south to north. we...
  25. Osvaldo

    Understanding magnetic balance in AC induction motors

    I need to find out how calculate magnetic forces appearing in AC induction motor rotor when it is not magnetically centered inside the stator.
  26. J

    Faraday's Unipolar Induction Equation

    Hello all, I have a question regarding Maxwell's Equations and Faraday's unipolar induction equation. If we study the case of a cylindrical magnet with a radius of r which is rotating about its axis with angular velocity w. The electrons within the magnet collide with the moving atoms, causing...
  27. Yousuf Ejazi

    Calculate the induced EMF for angle α

    I have a question in my book and it’s confusing me a bit. I tried to search online for similar solved problems but couldn’t succeed. So here it goes:Calculate the induced EMF in a conductor loop when the angle between \vec{A} ~ and~ \vec{B} ~is~ changed ~from ~{0 °}~ to ~{α °} in 1 second...
  28. N

    Induction: Electromagnet and Aluminium Ring

    I need some help with this task, I am stuck. 1. Homework Statement An aluminium ring is placed on top of an elctromagnet. a. The electro magnet is connected to AC. What will happen with the aluminium ring? Motivate your answer. b. What will happen with the ring if the electromagnet is...
  29. S

    MHB How to prove by Induction on k. using Fibonacci Numbers

    Please guide me how to prove the last equation by induction. Regards! Conversely, any representation of the form (Zeckendorf's Theorem) $$n=F_{k_{1}}+F_{k_{2}}+...+F_{k_{r}}, \quad k_{1}\gg k_{2}\gg ... \gg k_{r} \gg 0.$$ Implies that $$ F_{k_{1}} \leq n < F_{k_{1}+1}, $$ because the largest...
  30. G

    Sign of the Faraday-Lenz equation

    Homework Statement Hello, good afternoon, I have a question that's coming up when I'm doing electromagnetic induction exercises. In some exercises they ask me to determine the electromotive force (iee), and when they do the exercise resolution they do it with absolute value. In other...
  31. nomadreid

    I Hilbert's omega rule, induction, omega-consistency

    As I understand Hilbert's omega rule for a first-order proposition P over the natural numbers, P(0) &P(1) &P(2) &... ⇒ ∀n∈ℕ P(n) :star: which seems to be the same as ω-consistency. Is there a difference? Further, the axiom schema of induction has each axiom for a proposition P over the...
  32. L

    Engineering How Does Reducing Excitation Affect a Synchronous Induction Motor's Performance?

    Homework Statement (i) A star-connected, three-phase synchronous induction motor takes a current of 10 amps from a 415 volt supply at unity power factor when supplying a steady load. If the synchronous reactance is 5 ohms/phase and the resistance is negligible, draw the phasor diagram and...
  33. EF17xx

    Electromagnetic induction emf question

    Homework Statement The correct answer is D Homework Equations The Attempt at a Solution The current is clockwise( in the first picture at least) due to right hand rule. The magnetic flux is changing, the direction of the induced emf depends on the direction of change (increase or...
  34. G

    Curious question regarding Faraday disc / induction, etc.

    before I ask anything I know this has been probably talked about a lot in the past from various different angles, yet when I am searching for answers I cannot find a satisfactory answer that would look at it from the perspective I wish so here is my try. I've talked with a mentor of mine about...
  35. MermaidWonders

    MHB What is the current in a wire loop outside a solenoid with a changing current?

    A 2000-turn solenoid is 2.0 m long and 15 cm in diameter. The solenoid current is increasing at 1.0 kA/s. (a) Find the current in the 10-cm-diameter wire loop with resistance 5.0 $\varOmega$ lying inside the solenoid and perpendicular to the solenoid axis. (b) Repeat for a similarly-oriented...
  36. A

    Induction Motor to Stepper Motor

    Can induction motor be used as a stepper motor ?
  37. ertagon2

    MHB Induction and Polynomials

    Can someone check if my answers are right and help me with the missing ones.
  38. C

    MHB Solve Sigma 1/(k(k+1)): Step-by-Step Guide

    Use the fact that \frac{1}{k} - \frac{1}{k+1} = \frac{1}{k(k+1)} to show that n sigma (\frac{1}{k(k+1)}) = 1- \frac{1}{n+1} r=1 What do I need to do to solve it?
  39. Shadow89

    Optimal values for power in an induction heater?

    I really want to build a simple but powerful induction heater for hobby blacksmithing. Metal shaping and the design of the cooling system within the coils are not a problem. I need help understanding the electronics Obviously I must first use a transformer as a safety against "backlash" to the...
  40. N

    Mathematical Induction with Sigma notation

    Prove by mathematical induction that n sigma r^3 = n^2(n+1)^2/4 r = 1 so far I have 1 sigma r^3 = 1^2(1+1)^2/2 r=1 1 = 1(4)/2 1 = 4/2 1 = 2 I'm not sure what to do after this for the k+1 case.
  41. negut constantin

    Weber induction force and EMF

    We know that primary of a transformer is a coil(mono or multilayer) from a conductive material, usually copper, which have a proper lattice.Inside the lattice is a cloud of free electrons .When in circuit, a voltage is applied at the coil heads. 1.When voltage is applied , do we have a...
  42. T

    MHB How would I prove this by induction?

    How would I prove this by induction? $$\sum_{k=0}^n\frac{n!}{n^kk!(n-k)!}=\frac{(n+1)^n}{n^n}$$
  43. Z

    Finding the Torque vs RPM curve for a 3-phase AC induction motor

    Dear community, For one of the projects which I am currently working on, I want to quantify my three phase AC induction motor in the following ways: 1. Find Torque vs RPM Curve 2.Find Efficiency vs RPM Curve Now since I don't have the industrial "Torque Sensor" to plot Torque vs RPM curves, I...
  44. Jucimar

    Can self induction increase power consumption?

    Hi folks, My question relays on the difference of consumption power of a coil when it's on AC and CC regime. A coild is made of copper wire with air core, aplied on it a CC current of 1 amp, a measure was taked and nottice a 5W of power consumption. Then the power supply is changed to and AC...
  45. ?

    EM fields of two opposite moving point charges

    I wrote and solved this problem but am having serious doubts about the answer I obtained. Homework Statement Two point charges \pm q move along the z-axis with velocity \pm v. If they are at the origin when t=0, what is the electric field magnitude a distance r from the z-axis? Homework...
  46. K

    Why the voltage on the primary coil changes in an induction stoveplate?

    Translated from Norwegian high-school physics book: I wonder how and why does the voltage change on the primary coil when the cookware is removed or boiled over?
  47. Cardinalmont

    I Short Circuiting the Secondary of a Transformer/Self Impedence

    Thank you for reading my post. I was watching the video this video on transformers and saw an amazing demo: From 4:00-5:20 the guy does an experiment where he has a simple transformer circuit set up: On the primary side of the transformer he has an AC power supply, and a lightbulb. On the...
  48. Humbleness

    Electromagnetic Induction Problem (Very Introductory)

    Homework Statement 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 reversed - explain...
  49. 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...
  50. 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...
Back
Top