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Bijection between open interval and half-open interval

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Please help with the following problem. Provide step by step calculations.

Prove that there is a bijection from the open interval (0, 1) to the half-open interval (0, 1].

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Solution Summary

The following problem helps with a linear algebra problem. A function f:(0, 1) -> (0, 1] is provided, along with a detailed proof that f is not only a one-to-one function but also an onto function. Step by step calculations are given.

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We will define a function f:(0, 1) -> (0, 1] such that f is one-to-one and onto.

Note that 1 is not an element of the open interval (0, 1), but that 1 is an element of the half-open interval (0, 1].

Moreover, every element of (0, 1) is also an element of (0, 1].

Thus the main challenge is to come up with a way to map some element of the open interval (0, 1) to 1 and still be sure that, for every element y of (0, 1], there is some element x of (0, 1) such that f(x) = y. (This property will ensure that the mapping is onto.)

Besides that, of course, we need to be sure that for every element y of (0, 1], there is ...

Solution provided by:
Georgia Martin, PhD
Education
  • AB, Hood College
  • PhD, The Catholic University of America
  • PhD, The University of Maryland at College Park
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