Question

Prove that the set \(A=\\{(m, n) \in \mathbb{N} \times \mathbb{N}: m \leq n\\}\) is countably infinite.

Short answer

The set \(A = \{(m,n) \in \mathbb{N} \times \mathbb{N}: m \leq n\}\) is countably infinite. This is shown by establishing a bijective function \(f: A \rightarrow \mathbb{N}\) as \(f(m,n) = \frac{n(n + 1)}{2} + m\), which correlates each pair \((m, n)\) in \(A\) with a unique natural number. Thus, there exists a one-to-one correlation between \(A\) and the natural numbers, proving the countability of \(A\).

Step-by-step solution

Identify the Set \(A\)

We first identify our set \(A\) to be all pairs \((m, n)\) in \( \mathbb{N} \times \mathbb{N} : m \leq n\), i.e., the set of all pairs of natural numbers where the first element is less than or equal to the second.

Understanding the Concept of Countability

A set is said to be countably infinite if there exists a function that correlates each of its elements with a unique natural number, i.e., if there exists a bijective correlation between the set and the natural numbers. We need to find such a function for our set \(A\).

Establishing a Bijective Function

We define our function \(f: A \rightarrow \mathbb{N}\) as \(f(m,n) = \frac{n(n + 1)}{2} + m\). It can be noted that this function does correlate each pair \((m, n)\) with a unique natural number. If \((m_1, n_1) \neq (m_2, n_2)\), then \(f(m_1, n_1) \neq f(m_2, n_2)\), ensuring the uniqueness of our function.

Proving Countability

Having established our function \(f: A \rightarrow \mathbb{N}\) as bijective, we thus have a one-to-one correlation between our set \(A\) and the set of natural numbers. Therefore, we can conclude that our set \(A\) is countably infinite.

Key concepts

Bijective Function

When we explore the concept of a bijective function, we encounter a foundational piece of set theory and mathematical analysis. A function is considered bijective if it is both injective (one-to-one) and surjective (onto). In simpler terms, this means every element in the domain of the function is paired with a unique element in the codomain, and every element in the codomain is connected to some element in the domain.

Within the context of the exercise, establishing a bijective function between the set of ordered natural number pairs \( (m, n) \) and the natural numbers themselves \( \mathbb{N} \) is crucial for proving countability. The proposed function \( f(m,n) = \frac{n(n + 1)}{2} + m \) is a classic example, often employed in proofs involving countable sets.

Considering the exercise improvement advice to make these concepts clearer, it is helpful to visualize or provide examples. For instance, applying the function to a small pair of numbers helps illustrate the one-to-one correspondence. If we take \(m=2\) and \(n=3\), by our function we’d get \( f(2, 3) = \frac{3(3 + 1)}{2} + 2 = 8 \), denoting a unique image in the natural numbers, which underscores the function's bijectivity.

Natural Numbers

The natural numbers are the simplest numerical constructs that we begin learning as children, usually starting from one and continuing indefinitely. These numbers are symbolized by the set \( \mathbb{N} \) and include \(1, 2, 3, ...\), excluding zero by traditional convention, though some modern definitions include it.Understanding natural numbers is critical in fields such as arithmetic, algebra, and set theory. They are used to count and order items which is why they are often referred to as 'counting numbers'. In our exercise, these numbers form the domain and codomain for demonstrating the countability of a set. Since they have a natural order from least to greatest, it is possible to arrange any countable set in a sequence that can be listed in a similar fashion, a concept central to proving countability.

Set Theory

Moving onto set theory, which is the study of sets or collections of objects. It serves as the foundation for most of mathematics. In the realm of set theory, we classify different types of sets based on their size or 'cardinality'. A set can be finite, countably infinite, or uncountably infinite. This categorization helps us understand the nature of mathematical infinity and the structure of number systems.

In our case, we are interested in a set that can be listed or 'counted' one by one, which is why we refer to it as 'countably infinite'. For a set to have this property, we must be able to match each element to a distinct natural number. Referring back to our exercise, showing that the set \(A\) is countably infinite, involves constructing a systematic method (a bijective function), which proves a one-to-one correspondence between the set and the natural numbers. Additionally, by referring to the step-by-step solution, breaking down complex definitions into more digestible examples or explaining them using familiar terms can significantly enhance understanding.