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How can I find an injective function from $h ∈ (\mathbb N → {0, 1}) → A$. $A = \{f : N → Z|∀n, m ∈ N.(n > m) → (f(n) < f (m))\}$?

I was trying to define it as follows:

$h=-n-f(n)$, but I know that doesn't work since it only maintains $f(n) \leq f (m)$ but not $f(n) < f (m)$.

Would love to hear your thoughts.

Moshe King
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  • Please write in MathJax – A. Goodier Dec 22 '17 at 21:54
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    So, you are asking for an injection of the set of binary sequences to the set of strictly decreasing integer sequences. Let $(a_1,a_2,a_3,\dots)$ be a binary sequence. Let us define an integer sequence $(f_1,f_2,f_3,\dots)$ recursively in a useful way. We want to make sure that $f_1>f_2>f_3>\dots$ but we also want it to be unique according to our sequence $(a_1,a_2,a_3,\dots)$... how about if we decrease by one or we decrease by two depending on ______ and we can write $f_n$ then as ______ and we can prove that this mapping is injective by _____ and that $f$ satisfies our properties. – JMoravitz Dec 22 '17 at 23:35
  • What does h ∈ (N → {0, 1}) → A mean? – William Elliot Dec 23 '17 at 01:35
  • duplicate question - https://math.stackexchange.com/q/2576603/432081 – CopyPasteIt Dec 23 '17 at 04:42

1 Answers1

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Given a binary sequence, a=(a1, a2, a3,....) we define h(a)=f where f is defined on the set of natural numbers as follows; f(1)= a1 and for k>=1 let f(k+1)=f(k)-ak-1.The function f is decreasing and h is a one-to-one association. For example the sequence 1,0,0,1,1,1,0,0,1,.... maps to the sequence 1,0,-1,-3,-5,-7,-8,-9,-11,... Note that the map h is one-to-one because a1=f(1) and for k>=2, ak=f(k)-f(k+1)-1 defines the inverse of h.

Mohammad Riazi-Kermani
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