Inductive Proof.

  • Thread starter cragar
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  • #1
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Homework Statement


Let [itex] n_1=min(n\in\mathbb{N}:f(n){\in}A) [/itex]
As a start to a defintion of g:N→A, set [itex]g(1)=f(n_1) [/itex]
Show how to inductively continue this process to produce a 1-1 function g from
N onto A.

The Attempt at a Solution


[itex] g(1)=f(n_1) [/itex] so this is our base case for induction.
so [itex] g(2)=f(n_1+1) [/itex]
If I understand this correctly g is a function that has input values of natural numbers and maps these to the set A.
So I guess I need to show that f(n) is in A and f(n+1) is in A
By definition f(n) is in A for all n, so f(n+1) is in A for all n.
Could I maybe do a proof by contradiction and assume that f(n+1) was not in A and show that it was because n+1 is in the Naturals, therefore it works for f(n), and f(n+1)
 
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Answers and Replies

  • #2
jbunniii
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Are there some additional assumptions that you haven't listed? It is obviously not going to be possible to construct a bijection between [itex]\mathbb{N}[/itex] and [itex]A[/itex] unless [itex]A[/itex] is countable. Also, there must be some assumption about [itex]f[/itex]. In general, the set [itex]\{n \in \mathbb{N} : f(n) \in A\}[/itex] could be empty, in which case the minimum isn't even defined.
 
  • #3
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ok, ya sorry, It says there exists [itex] f:\mathbb{N} --->B [/itex] which is 1-1 and onto.
Let [itex] A\subseteq B [/itex] be an infinite subset of B.We must show that A is countable. The question was phrased in 2 parts and I thought they were separate.
Thanks for looking at that.
 

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