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Chapter 3: Problem 47

Prove each. The set \(Q^{+}\) of positive rational numbers is countable.

Short Answer

Expert verified
To prove that the set of positive rational numbers, \(Q^+\), is countable, construct a table of positive rational numbers, remove duplicate entries and reduce all fractions, then map the table to natural numbers diagonally. Define a one-to-one correspondence, \(f: \mathbb{N} \to Q^{+}\), where \(f(n)\) returns the \(n\)-th positive rational number in the sequence. Since \(f\) assigns each positive integer to a unique positive rational number, the set \(Q^+\) is countable.

Step by step solution

01

Construct a table of positive rational numbers

Create a table where the rows and columns are indexed by positive integers. In each cell \((m, n)\) of the table, put the positive rational number \(\frac{m}{n}\).
02

Exclude duplicate elements and reduce all fractions

As there may be fractions that are equivalent within the table (e.g., \(2/4\) and \(1/2\)), remove all duplicate entries by only including fractions in their reduced form in the table. To do this, exclude cells with \(\frac{m}{n}\) for which \(m\) and \(n\) share a common divisor greater than 1.
03

Map the table to natural numbers

We can map the cells of the table to natural numbers by moving diagonally through the table, starting from the upper-left corner and following these diagonals in the following pattern: \(\frac{1}{1} \to \frac{2}{1} \to \frac{1}{2} \to \frac{1}{3} \to \frac{2}{2} \to \frac{3}{1} \to \frac{4}{1}\), and so on. In each step, move along the decreasing values of the row indices and increasing values of the column indices. This gives us a sequence of positive rational numbers that contains each distinct reduced rational number exactly once.
04

Define a one-to-one correspondence

Now, we can define a one-to-one correspondence, \(f: \mathbb{N} \to Q^{+}\), where \(f(n)\) returns the \(n\)-th positive rational number in the sequence created in Step 3. This function assigns each positive integer to a unique positive rational number.
05

Conclude that \(Q^{+}\) is countable

Since we have constructed a function, \(f\), that provides a one-to-one correspondence between the set of natural numbers and the set of positive rational numbers, we can conclude that the set of positive rational numbers, \(Q^{+}\), is countable.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Positive Rational Numbers
Positive rational numbers form a fundamental concept in mathematics. These numbers can be expressed as a fraction \( \frac{p}{q} \), where \( p \) and \( q \) are integers, and \( q > 0 \). Key characteristics include:
  • They are positive, meaning both \( p \) and \( q \) are greater than zero when \( p = 0 \), the number itself becomes zero, which is excluded.
  • They can represent both whole numbers and non-integers, giving a flexible way to handle division of whole numbers.
  • Every positive rational number can be expressed in its simplest form, by dividing both numerator and denominator by their greatest common divisor.
Understanding positive rational numbers is crucial as it allows us to measure and express proportions, ratios, and understand division in a more versatile manner.
These numbers play a key role in forming the link between integers and real numbers, encapsulating the concept of "parts of a whole."
One-to-One Correspondence
In mathematics, one-to-one correspondence refers to a bijective function, where each element of one set is paired uniquely with an element of another set.
For our exercise, the one-to-one correspondence is vital in proving countability:
  • We establish that each natural number maps to a unique positive rational number.
  • This mapping ensures no positive rational number is left out, and every one is paired with a different natural number.
This unique mapping is crucial for demonstrating that the set of positive rational numbers can be linked directly to the set of natural numbers in a way that every member of both sets is accounted for.
It showcases the systematic approach in organizing numbers and aids in understanding properties of both countable and uncountable sets.
Natural Numbers
Natural numbers are the simplest form of numbers used in mathematics.
Represented as \( \{1, 2, 3, \ldots\} \), these numbers are used primarily for counting and ordering.
Here are some essential features of natural numbers:
  • They start from one and go on infinitely.
  • No fractions, decimals, or negative numbers are included.
In the context of the exercise, natural numbers serve as the "index" or "address" for the positive rational numbers.
By mapping each positive rational number to a natural number, we facilitate an easy way to "list" or "order" the positive rational numbers.
This systematic approach using natural numbers helps to demonstrate their role in the concept of countability and creating a manageable structure for larger mathematical sets.
Mathematical Proof
A mathematical proof is a rigorous method of establishing the truth of a statement or theorem.
Proofs employ a logical sequence of statements and justifications:
  • The proof structure typically begins with an assumption or initial condition.
  • This follows through with a series of logical deductions, which rely on previously known and accepted truths.
  • Concludes by showing that the initial assumption leads inevitably to the desired conclusion.
In proving the countability of the set of positive rational numbers, the construction of a one-to-one correspondence serves as a major step in this logical process.
This proof involves recognizing patterns, using logical sequences, and testing these sequences against known rules or axioms in number theory.
Understanding mathematical proofs aids in developing critical thinking and problem-solving skills, as they train mathematicians to argue systematically and avoid assumptions without evidence.

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