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Consider the set $S \subset \mathbb{N}^2$ of ordered pairs of integers defined by the following recursive definition:

• $(3, 2) \in S$ (basis)

• If $(x, y) \in S$, then $(3x − 2y, x) \in S$ (recursive step)

Also consider the set $S' \subset \mathbb{N}^2$ with the following non-recursive definition:

$$ S' = \{(2^{k+1} + 1, 2^k + 1) \mid k \in \mathbb{N}\}. $$

Prove using structural induction that $S \subseteq S'$.

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  • $\begingroup$ That's false.. by your definition, $S$ may contain also (0,0) $\endgroup$
    – Exodd
    Sep 27, 2014 at 19:43
  • $\begingroup$ It's easy to prove the opposite: $S'\subseteq S$ $\endgroup$
    – Exodd
    Sep 27, 2014 at 19:45
  • $\begingroup$ how would I go about doing that? Don't I used complete induction to prove the opposite? It says I need to use structural induction. $\endgroup$ Sep 27, 2014 at 19:46
  • $\begingroup$ both ways are the same (if I get what you want): (3,2) is in the form of the element of $S'$, with $k=0$, but then the recursive step transform it in (5,3) that is also an element of $S'$ with $k=1$. Now, by induction, you shw that if $(x,y)=(2^{k+1}+1,2^k+1)$, then $(3x-2y,x)=(2^{k+2}+1,2^{k+1}+1)$ $\endgroup$
    – Exodd
    Sep 27, 2014 at 19:50
  • $\begingroup$ how would i go about showing it? like this: 3x - 2y = 2^(k+2)+1 and x = 2^(k+1)+1 so then 3(2^(k+1)+1)-2y=2^(k+2)+1 and I solve for y. I tried that came up with no conclustions @Exodd $\endgroup$ Sep 27, 2014 at 20:10

1 Answer 1

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I'm assuming that $S$ is the smallest set containing $(3,2)$, and for each $(x,y) \in S$ also $(3x-2y,x) \in S$.

Hint: If $(x,y) \in S$ then for some $t \geq 0$, $(x,y)$ is obtained by $k$ applications of the recursive step. Use induction on $k$ to show that $(x,y) = (2^{k+1}+1,2^k+1)$. In the same way you can also prove the reverse inclusion, and conclude that $S = S'$.

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  • $\begingroup$ Thank you!! I proved it this way. But the question says to prove using both structural and simple/complete induction. I feel I just proved it using simple/complete induction. How is structural suppposed to be different from this? For my proof k=1 and i proved 3x+2y = 2^(k+1)+1 $\endgroup$ Sep 27, 2014 at 20:42
  • $\begingroup$ The question is stupid. Structural induction in this case degenerates to simple induction. You should ask your instructor what they mean by the magic words "structural" vs. "simple". I suspect that is some pattern that they expect you to fill when writing up the proof which is slightly different in both cases. You can tell them from me that I think this is silly. Nobody should think this way. The idea of math is to identify common patterns, not to separate them in some arbitrary manner. $\endgroup$ Sep 27, 2014 at 20:45

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