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ssamsymn
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Can you help me to construct a 1-1 mapping from real numbers onto non-integers? thanks
jbunniii said:Let ##(z_n)## be an enumeration of the integers, and let ##(r_n)## be any sequence of distinct non-integers (##n = 0, 1, 2, \ldots##). Define the map ##f : \mathbb{R} \rightarrow \mathbb{R}\setminus \mathbb{Z}## by
$$f(r_n) = r_{2n}$$
$$f(z_n) = r_{2n+1}$$
$$f(x) = x \textrm{ for all other }x$$
It's easy to see that this is a bijection.
micromass said:You want a bijection [itex]f:\mathbb{R}\rightarrow \mathbb{R}\setminus \mathbb{Z}[/itex]??
Now, it is not so hard to see that such a bijection has to exist. So if you're only interested in bijection and not existence, then you basically only have to apply the wonderful Cantor-Bernstein-Shroder theorem.
If you want an actual map, then this requires a bit more work. It might be easier to first find a bijection [itex]f:\mathbb{R}\rightarrow \mathbb{R}\setminus \{0\}[/itex] and then apply the same trick on your problem. For the easier problem, the trick is to select a sequence in [itex]\mathbb{R}[/itex] such as [itex]x_n=n[/itex] for [itex]n\geq 0[/itex]. Then we can define
[tex]f(x)=x~\text{if}~x\neq x_n~ \text{and}~f(x_n) = x_{n+1}[/tex]
Do you see that that works?? Can you apply this same idea on your problem?
If you want something specific, just take something like ##r_n = (2n+1)/2##, i.e., the sequence ##1/2, 3/2, 5/2, \ldots##. Any sequence of real numbers will work as long as they are all distinct and none of them are integers. Drawing a picture might help if it's not clear what the map is doing.ssamsymn said:thank you very much.
I don't fully understant how I can pick real numbers with n indices.
I am not familiar with the notation, I need a valid function to show this map is one to one and onto.
jbunniii said:If you want something specific, just take something like ##r_n = (2n+1)/2##, i.e., the sequence ##1/2, 3/2, 5/2, \ldots##. Any sequence of real numbers will work as long as they are all distinct and none of them are integers. Drawing a picture might help if it's not clear what the map is doing.
A map from real numbers to non-integers is a function that assigns a non-integer value to each real number input. It can also be called a transformation or a mapping.
A map from real numbers to non-integers can be useful in many mathematical and scientific applications, such as representing non-integer quantities like probabilities, measurements, or approximations.
A map from real numbers to non-integers is a specific type of function that only deals with real numbers as inputs and outputs non-integer values. Regular functions can have inputs and outputs from a wider range of numbers, including integers and complex numbers.
Yes, a map from real numbers to non-integers can be represented graphically using a graph or a curve. The x-axis will represent the real numbers, and the y-axis will represent the non-integer values.
The mathematical notation for a map from real numbers to non-integers is often written as f: R → N, where R represents the set of real numbers and N represents the set of non-integer numbers.