Constructing a Bijection Between an Interval and the Set of Real Numbers

In mathematics, particularly in real analysis, constructing a bijection between an interval and the set of real numbers is a key concept. This article explains the step-by-step process of creating such a bijection, focusing on the interval [-sqrt{21}] and extending it to the real numbers mathbb{R}

Introduction to Bijection

A bijection is a function between two sets that is both one-to-one (injective) and onto (surjective). In simpler terms, it means that each element of the first set is paired with exactly one element of the second set, and vice versa.

Step-by-Step Process

Step 1: Mapping the Interval to a Standard Interval

First, we need to map the interval [-sqrt{21}] to a standard interval like [0, 1]. We define a linear transformation to achieve this:

Calculate the length of the interval: n Text{Length} 1 - (-sqrt{21}) 1 sqrt{21}

The transformation function can be given as:

gy frac{x - (-sqrt{21})}{1 sqrt{21}}

This function shifts the interval to the right by sqrt{21} and then scales it to fit within [0, 1].

Step 2: Mapping the Standard Interval to mathbb{R}

Next, we need to map the interval [0, 1] to the set of real numbers mathbb{R}. A common bijection from [0, 1] to mathbb{R} is the tangent function:

hx tan{left(pi y - frac{pi}{2}right)}

This function maps 0 to -infty and 1 to infty.

Step 3: Combining the Functions

The combined function that provides a bijection from [-sqrt{21}] to mathbb{R} is:

fx tan{left(pi left(frac{x - (-sqrt{21})}{1 sqrt{21}}right) - frac{pi}{2}right)}

This function is both injective (one-to-one) and surjective (onto), thus establishing a bijection between the two sets.

Alternative Method: Piecewise Continuous Mapping

An alternative method involves a piecewise continuous mapping. This method can be described as follows:

Take the rightmost half of the interval [0, 1] excluding the middle point and reverse it. This can be defined by the function: n fx frac{3}{2} - x text{ for } x in [frac{1}{2}, 1]

This transformation covers the interval [frac{1}{2}, 1] and maps it back to itself with a reflection. The process is repeated for smaller and smaller subintervals until the entire interval is covered. The final transformation can be expressed as:

x to 3 cdot 2^{lceillog_2 xrceil - 1} - x

This piecewise continuous function ensures that every point in [0, 1) is mapped uniquely to a point in [0, 1].

Conclusion

In conclusion, constructing a bijection between an interval and the set of real numbers is a powerful tool in real analysis. The methods described here, whether through linear transformations or piecewise continuous mappings, provide a clear and effective way to achieve this.

For further exploration, the reader is encouraged to experiment with different intervals and functions to understand the intricacies of these mappings.