Question

Show that each countable, ordered set can be embedded in the rationals.

Short answer

Any countable ordered set can be embedded into \( \mathbb{Q} \) by utilizing its density to map order-preservingly.

Step-by-step solution

Understand the Problem

We need to demonstrate that every countable ordered set can be represented within the set of rational numbers \( \mathbb{Q} \). This means finding an order-preserving function from the countable set into \( \mathbb{Q} \).

Define Terms and Concepts

A countable set is one that has the same cardinality as the set of natural numbers \( \mathbb{N} \). An ordered set has a binary relation that determines the order of its elements. A function is order-preserving if \( a < b \) implies \( f(a) < f(b) \), where \( f \) is the function and \( a, b \) are elements of the ordered set.

Approach the Solution

Since the given set is countable, we can list its elements as \( a_1, a_2, a_3, \ldots \). We need to assign rational numbers to these elements in such a way that their natural order is preserved.

Use Rational Density

The key property of rational numbers \( \mathbb{Q} \) is their density: between any two rational numbers, there exists another rational number. This property allows us to insert elements of the countable set into \( \mathbb{Q} \) while preserving order.

Construct the Embedding Function

Start by mapping the first element \( a_1 \) to any rational number, say \( r_1 \). For each subsequent element \( a_i \), choose a rational number \( r_i \) such that it maintains the order: if \( a_i < a_j \), then ensure \( r_i < r_j \). This is possible due to the density of rational numbers.

Validate the Embedding

Verify that the function you've created preserves the order: for every pair of elements \( a_i, a_j \) in the ordered set, if \( a_i < a_j \), our mapping satisfies \( f(a_i) = r_i < r_j = f(a_j) \). Because we've used the density property, this is achievable for any countable list of elements.

Conclusion

Since we've managed to map a countable sequence while preserving order, we've successfully embedded the countable ordered set into the rationals.

Key concepts

Rational Numbers

Rational numbers are numbers that can be expressed as the quotient of two integers, where the denominator is not zero. They are denoted by \( \mathbb{Q} \), indicating the set of all such numbers. For example, \( \frac{1}{2}, \frac{-3}{4}, \) and \( 5 \) are all rational numbers.
A key property of rational numbers is their density. This means that between any two rational numbers, there exists another rational number. This property is extremely powerful, especially when working with ordered sets. To explain further:

• Between any two rational numbers \( a \) and \( b \), you can find another rational number \( c \) such that \( a < c < b \).
• This is why rational numbers can be used so effectively in embedding countable sets while preserving order. It allows for flexibility in assigning values to elements in an ordered set.

The rational numbers are also countable, which means they can be put into one-to-one correspondence with the natural numbers. This is important when embedding ordered sets into \( \mathbb{Q} \).

Order-Preserving Function

An order-preserving function, sometimes called a monotonic function, maintains the order of elements between sets. If you have two elements \( a \) and \( b \) in an ordered set, and \( a < b \), an order-preserving function \( f \) will ensure that \( f(a) < f(b) \) in the target set as well.
This is crucial when embedding a countable ordered set into the rational numbers. The function must:

• Map each element of the ordered set to a rational number.
• Preserve the existing order of the elements.
• Take advantage of the density of \( \mathbb{Q} \) to ensure no two mappings conflict.

Order-preservation is necessary for the integrity of the original set to remain intact within the new set. This way, relative sizes and positions in the list remain true to their original ordering.

Set Theory

Set theory is the branch of mathematical logic that studies sets, which are collections of objects. It's the foundation for most of mathematics and essential for understanding many concepts, including embedding and cardinality.
Key elements of set theory include:

• Countable sets: These are sets whose elements can be put into a one-to-one correspondence with the natural numbers, like \( \mathbb{N} \) and \( \mathbb{Q} \).
• Ordered sets: These sets have a specific sequence where elements are arranged in a defined order, usually indicated by \( < \) or \( \leq \).
• Relations and mappings: These concepts help define how elements in different sets relate, vital for functions like embeddings.

Understanding these concepts is crucial in discussing how countable ordered sets can be embedded into rational numbers using order-preserving functions.

Cardinality of Sets

Cardinality refers to the number of elements in a set and is a fundamental concept in set theory. For finite sets, cardinality is simply the count of distinct elements. For infinite sets, it's more complex, as we compare the "sizes" of infinite sets by establishing one-to-one correspondences.
In the context of our problem:

• Countable sets: Sets that have the same cardinality as \( \mathbb{N} \), like the rational numbers, are termed countable. This means they can be listed in an infinite sequence where each element corresponds to a natural number.
• The concept of cardinals helps us to understand and prove that any countable ordered set can potentially be embedded into the rational numbers without altering its order.

Knowing a set's cardinality helps us manage it within our mathematically structured world, ensuring that operations like embedding are feasible and logically sound.