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I have to find the number of subsets within the set 1, 2, 3, ....n that contain no two consecutive integers, where 0 and n also count as consecutive.

I have tried this proof where 0 and n do not count as consecutive, and I know it roughly follows the Fibonacci sequence, but it is the 0 and n part that is confusing me. Does this mean that the empty set and n are consecutive? And therefore the set containing just n is consecutive? I have seen questions like this where 1 and n count as consecutive, but the fact that I am asked to do this with 0 and n is confusing to me.

The more detail in explaining your answer the better. Thank you.

SBlack412
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  • I think this must be a mistake $-$ the question meant to say that $1$ and $n$ count as consecutive. – TonyK Oct 02 '16 at 18:20
  • If you meant to say that $1$ and $n$ count as consecutive, the earlier question is an exact duplicate. If the set was supposed to be ${0,1,\ldots,n}$, this is a very minor variation on the earlier question, whose answer requires only very minimal adjustment. – Brian M. Scott Oct 02 '16 at 20:07
  • The question I am tasked with answering says the set from {1, 2, ..., n} and does ask about 0 and n specifically being consecutive, but this might have been a typo? That is why I felt that the duplicate question didn't answer my question because of the confusing way that mine is worded. – SBlack412 Oct 03 '16 at 01:12

1 Answers1

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I assume that you here mean the set $\{ 0,1,2,3, \ldots, n\}$, since you refer to the number $0$.

The condition of being consecutive means that the successor relation is circular -- $1$ and $2$ are consecutive integers, but $n$ and $0$ are also consecutive integers. Consider as an example the universe $\{ 0,1,2,3,4 \}$. Here $\{1,3\}$ is a subset that does not contain any consecutive integers, whereas $\{3,4\}$ and $\{4,0\}$ are subsets that contain consecutive integers.

To find $S(n)$, the number of subsets of $\{ 0, \ldots, n\}$ that do not contain any consecutive integers, the most natural approach is to write down a recursive definition of $S(n)$.

Hint: A subset of $\{ 0, \ldots, n\}$ that does not contain any consecutive integers can be either a set of this kind chosen from the first $\{ 0, \ldots, n-1 \} $ or one that contains neither $0$ nor $n$.

Hans Hüttel
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