Bijection from Reals to [0,1]

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In summary, the conversation discusses the difficulty of mapping all real numbers into the closed interval [0,1]. The speaker suggests using a bijective function, such as the tangent function, but acknowledges that it becomes more challenging when considering the end points 0 and 1. They suggest using a piecewise function or mapping (0,1) to [0,1] or vice versa to simplify the problem.
  • #1
gravenewworld
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Hi, I'm trying to map all the reals into the interval [0,1]. I figured out that you can map all the numbers in the open interval (0,1) to all the reals by the function tan(pi(x-.5)) (so if I wanted a function from all the reals to (0,1) I could just take the inverse). But this problem is much harder when you consider the closed interval. Is there a way to modify the tangent function I gave to make another bijective function that hits the end points 0 and 1? Would I have to create some sort of peicewise function to remedy this problem?
 
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  • #2
The problem is you're naturally enough trying to thinking of continuous functions, as those are the ones we meet most.

Why not now map (0,1) to [0,1] or vice versa? It's not too hard once you start thinking of functions in the right way. For instance, let's send the numbers 1,1/2,1/3,1/4... to 1/2,1/3,1/4... How'd you do that? now define the function on [0,1] using that bit, adn sending x to x otherwise. What can you say about that map?
 
  • #3


Yes, you are correct that the function tan(pi(x-0.5)) maps all the numbers in the open interval (0,1) to all the reals. However, as you mentioned, this function does not include the endpoints 0 and 1. In order to create a bijective function that includes the endpoints as well, you can use a piecewise function.

One way to do this is by defining a function f(x) as follows:

f(x) = x for x ∈ (-∞, 0)

f(x) = tan(pi(x - 0.5)) for x ∈ (0, 1)

f(x) = 1 for x = 1

This function maps all the reals to the closed interval [0,1] in a bijective manner. The first part of the function maps all the negative numbers to 0, the second part maps the numbers in the open interval (0,1) to the reals using the tangent function, and the third part maps the number 1 to the endpoint 1.

In general, when dealing with intervals, it is important to consider the endpoints and include them in the function in order to have a bijective mapping. So, in this case, using a piecewise function is necessary to include the endpoints and create a bijection from the reals to the interval [0,1].
 

1. What is a bijection from Reals to [0,1]?

A bijection from Reals to [0,1] is a function that maps every real number to a unique number between 0 and 1. This means that each real number has one and only one corresponding value in the interval [0,1].

2. How is a bijection different from other types of functions?

A bijection is different from other types of functions because it is both injective (one-to-one) and surjective (onto). This means that each element in the domain (Reals) has a unique element in the range ([0,1]), and every element in the range has at least one corresponding element in the domain.

3. Why is it important to have a bijection from Reals to [0,1]?

Having a bijection from Reals to [0,1] is important because it allows us to easily convert between real numbers and numbers in the interval [0,1]. This is useful in many mathematical and scientific applications, such as in probability and statistics.

4. Is a bijection from Reals to [0,1] always possible?

Yes, a bijection from Reals to [0,1] is always possible. This is because both the set of real numbers and the interval [0,1] have infinite cardinality, meaning that they have the same number of elements. Therefore, it is always possible to find a one-to-one and onto mapping between them.

5. How do you prove that a function is a bijection from Reals to [0,1]?

To prove that a function is a bijection from Reals to [0,1], you must show that it is both injective and surjective. This can be done by setting up a proof for each property separately. For injectivity, you must show that if two real numbers have the same image in [0,1], then they must be the same real number. For surjectivity, you must show that every number in [0,1] has at least one corresponding real number in the domain. Once both properties are proven, the function can be confirmed to be a bijection from Reals to [0,1].

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