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How does mathematical induction actually work?

After surfing on the internet for a while, I found the following analogy. Consider rectangular tiles (dominoes) stacked on beside the other. When we force the first tile to fall, the others begin to fall. To actually know if all the tiles have fallen, we need to know the following-

  • If the first tile has fallen or not. (If not, non of them have fallen)
  • If the first tile has fallen, then we can pick some random tile from the stack to check if that has fallen. If this tile has fallen, then the previous tile must have also fallen.
  • From this, we can conclude that all the tiles have fallen.

But I don't understand how this idea of induction works with numbers?

Kaushik
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    Please explain better your doubts. This analogy is quite clear. Have you seen examples? – Danilo Gregorin Afonso Sep 12 '19 at 14:52
  • The first number is "good": Ok. If a number whatever is "Good", then the next number will be "Good". Therefore, every number is "Good". – Mauro ALLEGRANZA Sep 12 '19 at 14:52
  • @DaniloGregorin I have seen examples. In those examples they just use this method. My question is, 'How does this method work with numbers?' – Kaushik Sep 12 '19 at 14:53
  • Having shown that the first number is "Good", consider it for the second part of the proof and use it to show that the second number is "Good". Then use the second number... and so on. – Mauro ALLEGRANZA Sep 12 '19 at 14:54
  • I am sorry, I don't understand what you mean by the method working with numbers. – Danilo Gregorin Afonso Sep 12 '19 at 14:55
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    I think it's useful to realize that the natural numbers are usually defined inductively, allowing inductive proofs to almost piggy back on this construction. Obviously, this doesn't work for all types of construction, but this might build your intution. – Rushabh Mehta Sep 12 '19 at 14:56
  • @JoséCarlosSantos I saw that post. But I couldn't Understand. I'm Sorry. – Kaushik Sep 12 '19 at 14:56
  • @DonThousand Could you please elaborate? – Kaushik Sep 12 '19 at 15:00
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    That post has 9 answers. You read each of them and you understood none of them. Is that it? – José Carlos Santos Sep 12 '19 at 15:02
  • @JoséCarlosSantos I'm a High schooler and I never learnt about logarithm. So when I read the question over there, I lost hope of understanding it. So I asked it here expecting a simplified answer. – Kaushik Sep 12 '19 at 15:05
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    None of the nine answers mention logarithms. So, how come that's a problem? – José Carlos Santos Sep 12 '19 at 15:11
  • @JoséCarlosSantos Sorry. I'll go through that and if I understand, i'll close the thread by tomorrow. – Kaushik Sep 12 '19 at 15:34

1 Answers1

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If you just look at the definition, it may be hard to know. But if you really try it, it is easy to get it.

Let there is a statement $S\left(x\right)$, what induction works is:

$1)$ $S\left(1\right)$ is true

$2)$ If $S\left(k\right)$ is true, then $S\left(k+1\right)$ is true

Then, for all natural numbers $k$, $S\left(k\right)$ is true.

So we try to know the reason. Firstly, $S\left(1\right)$ is true from the first statement, then by the second statement, $S\left(2\right)$ is also true. Then also by the second statement, $S\left(3\right)$ is true. After that, you'll find that $S\left(4\right),S\left(5\right),S\left(6\right),\dots$ is also true. That's why induction works.

MafPrivate
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