Constructing a Bijective Function between Positive Integers and Rational Numbers

In mathematics, a bijective function, or a one-to-one correspondence, exists between two sets if every element of one set can be paired with exactly one element of the other set. In this article, we explore how to construct a bijective function from the set of positive integers (mathbb{N}) to the set of rational numbers (mathbb{Q}). This concept is crucial in understanding the cardinality of infinite sets and their enumerations.

Steps to Create the Bijective Function

One well-known method for constructing such a function involves listing the rational numbers in a systematic way. The rational numbers (mathbb{Q}) can be represented as fractions (frac{p}{q}) where (p) and (q) are integers, (q eq 0), and the fraction is in its simplest form, i.e., (gcd(p, q) 1).

Enumerating Rational Numbers

We can visualize the rational numbers in a two-dimensional grid where the rows correspond to the numerator (p) and the columns correspond to the denominator (q). This grid will include all positive fractions:

[Math] begin{array}{cccc} frac{1}{1} frac{1}{2} frac{1}{3} ldots hline frac{2}{1} frac{2}{2} frac{2}{3} ldots hline frac{3}{1} frac{3}{2} frac{3}{3} ldots vdots vdots vdots vdots end{array} [/Math]

Avoiding Duplicates

As we traverse the grid, we need to ensure that we only count each rational number once. This can be done using the property of being in simplest form. For instance, (frac{2}{2}) is not in its simplest form and should be omitted.

Defining the Function

We can define a function (f: mathbb{N} to mathbb{Q}) that assigns each positive integer to a unique rational number. One common method is to use the Cantor pairing function or a similar diagonal traversal that avoids duplicates.

Example of the Function

A specific way to define a bijective function (f) is as follows:

List the rational numbers (frac{p}{q}) in reduced form. Use a systematic approach to traverse the grid diagonally, omitting duplicates.

For example, the first few rational numbers in this enumeration would be:

(frac{1}{1}) (frac{1}{2}) (frac{2}{1}) (frac{3}{1}) (frac{1}{3}) (frac{2}{2}) not included since it is not in simplest form (frac{3}{2}) (frac{4}{1})

The function (fn) would yield the (n)-th rational number in this enumeration.

Conclusion

In summary, a bijective function from (mathbb{N}) to (mathbb{Q}) can be constructed by systematically listing the rational numbers in their simplest form using a diagonal argument. Each positive integer corresponds to a unique rational number, ensuring a one-to-one correspondence between the two sets.

This method not only demonstrates the fascinating interplay between infinite sets but also highlights the power of mathematical enumeration techniques. Understanding and applying such functions is essential for various fields, including number theory and computer science.