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For strong induction or "second induction" is it safe to assume for example when we assume it to hold true for some $k$ it also holds for $k-1$?

Bill Dubuque
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cheesewiz
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  • What is "second induction?" – Adam Rubinson Nov 03 '22 at 00:44
  • the book Im reading it refers it as the "second form of the principle of induction" I just called it second induction for short. – cheesewiz Nov 03 '22 at 00:46
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    You are going to have to spell out exactly what your book calls "the second form of the principle of induction." And you should never assume that your personal pet names will be understood by others. – Arturo Magidin Nov 03 '22 at 00:52
  • it is just strong induction... – cheesewiz Nov 03 '22 at 01:31
  • @ArturoMagidin I imagine a student learning about induction for the first time doesn't necessarily know when their textbook is employing "pet names" versus commonly accepted terminology. – AJY Nov 03 '22 at 02:14
  • @AJY. Read the comment. The book calls it "second principle of induction". The OP said "second induction", and that was the pet name: something they made up and used as if everyone knew what they were talking about. – Arturo Magidin Nov 03 '22 at 03:24
  • "Strong induction" can have different meanings, so again if you want to ask about specific hypothesis and whether you can use them or not, you need to be very specific about what statement you are talking about. For what I know as "strong induction", your question is nonsensical (you assume the result holds for all $k\lt n$, you want to prove it for $n$. So you never assume it "for some $k$". ) – Arturo Magidin Nov 03 '22 at 03:26
  • @ArturoMagidin "Any set of positive integers which contains the integers 1 and which contains k+1 whenever it contains the positive integers 1,2,3, . . . , k contains all positive integers" This is what I meant by the Second form of the principle of mathematical induction, I was under the impression the more common name for this principle was "strong induction" I assumed there was only 2 types of induction... – cheesewiz Nov 03 '22 at 03:34
  • Don't put it in comments.., put it in the post. – Arturo Magidin Nov 03 '22 at 03:50
  • Like I said, there are multiple ways of stating it. The statement I know as "strong induction for natural numbers" says "any set $S$ of natural numbers with the property that for all $n$, ( if every positive integer $k\lt n$ is in $S$, then $n\in S$) contains all positive integers." You'll notice the hypotheses are not identical to your phrasing. That means proofs using it will be slightly different. Which is why the precise phrasing is important if you are asking about what you can do when you use it in proofs. – Arturo Magidin Nov 03 '22 at 03:55
  • See here, and here, and here. – Arturo Magidin Nov 03 '22 at 04:04
  • alright I will be more precise next time thanks for the advice – cheesewiz Nov 03 '22 at 05:15

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Not always. If your base case is $k=1$, then when proving it for $k=2$, you cannot assume it's true for the $k=0$ case.

David Lui
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