Question

Show that the following sets are countably infinite: (a) \(\mid q \in \mathbb{Q}: q>10\\}\) (b) \(\left|q \in \mathbb{Q}: q^{2}

Short answer

All three sets are countably infinite, as they can be put in bijection with the natural numbers.

Step-by-step solution

Understand Countably Infinite Set

A set is countably infinite if it can be put into one-to-one correspondence with the set of natural numbers, \(\mathbb{N}\). This means that the elements of the set can be listed in a sequence: first, second, third, and so on.

Analyze Set (a)

For the set \(\{ q \in \mathbb{Q} \mid q > 10 \}\), consider that rational numbers \(q\) can be expressed as \(\frac{p}{r}\) where \(p\) and \(r\) are integers and \(r eq 0\). Since \(\mathbb{Q}\) itself is countably infinite and there is a bijection from the set of all rational numbers greater than 10 to the set of positive integers starting from the number 11, this set is also countably infinite.

Analyze Set (b)

For the set \(\{ q \in \mathbb{Q} \mid q^2 < q \}\), rearrange the inequality to indicate that \(q(q-1) < 0\). This implies \(0 < q < 1\). Since the interval \((0, 1)\) contains rational numbers and there is a known bijection between rational numbers in this interval and the natural numbers, this set is countably infinite.

Analyze Set (c)

For set \(\{ q \in \mathbb{Q} \mid q=\frac{i}{j} \text{ where } i \text{ is odd and } j \text{ is even} \}\), note that odd and even integers are subsets of integers, which are countably infinite. The pairs \((i, j)\), where \(i\) is odd and \(j\) is even, can be put into one-to-one correspondence with \(\mathbb{N}\) using a similar diagonalization argument used in enumerating rationals, showing this set is countably infinite.

Key concepts

Rational Numbers

Rational numbers are a fundamental part of mathematics. They are numbers that can be expressed as the quotient of two integers, i.e., \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b eq 0\). This means that rational numbers include whole numbers, fractions, and even repeating decimals.

A key property of rational numbers is that between any two rational numbers, there are infinitely many other rational numbers. This dense nature makes rational numbers a rich set to study within math, especially when dealing with set theory and number sequences. Understanding rational numbers helps in grasping more advanced concepts like real numbers and the structure of mathematical analysis.

Natural Numbers

Natural numbers are perhaps the simplest and among the first numbers we learn. They are the set of positive integers starting from 1, such as 1, 2, 3, and so on. These numbers are foundational in counting and ordering.

Mathematically, they serve as the building blocks for more complex number systems. Natural numbers are countably infinite, meaning that we can enumerate them in a sequence. This allows mathematicians to use natural numbers as a reference point when discussing the concept of infinity and countability in various sets, such as rational numbers.

Set Theory

Set theory is a branch of mathematical logic that deals with sets, which are collections of objects. These objects can be anything: numbers, letters, people, etc. The most basic operation is to consider whether a particular object is an element of a set or not.

Set theory provides a foundational framework for modern mathematics. It introduces concepts like

• countable and uncountable sets,
• relations between different sets,
• subsets and supersets,
• and operations like union, intersection, and set difference.

Understanding set theory is crucial when dealing with infinite sets and operations, which is common in higher mathematics.

One-to-One Correspondence

One-to-one correspondence, also known as a bijection, is a type of mapping between two sets. It indicates that each element of one set pairs exactly with one element of another set, and vice versa. This concept is crucial in determining the countability of a set.

To show that a set is countably infinite, you can find a bijection between the set in question and the natural numbers, \(\mathbb{N}\). This proves that the set can be "listed" in a sequence, much like listing the natural numbers: 1, 2, 3, etc.

Implementing one-to-one correspondence allows mathematicians to understand and compare the sizes of infinite sets, helping to distinguish between different types of infinity.