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Magazine Chapter 2: Problem 27
Show that the union of a countable number of countable sets is countable.
Short Answer
Expert verified
The union of a countable number of countable sets is countable, proven via a diagonal argument constructing a bijection.
Step by step solution
01
- Definition of Countable Sets
A set is countable if it is either finite or has the same cardinality as the set of natural numbers, meaning there exists a one-to-one correspondence with \(\backslashmathbb{N}\). Examples include natural numbers \(\mathbb{N}\), integers \(\mathbb{Z}\), and rational numbers \(\mathbb{Q}\).
02
- Definition of Union of Sets
Let's consider an infinite sequence of countable sets \(\backslash{A_1, A_2, A_3, \ldots\}\). The union of these sets is defined as \(A = \backslashbigcup_{i=1}^{\infty} A_i\), which includes all elements that are in any of the sets \(A_i\).
03
- Countable Union Principle
The claim is that the union of a countable number of countable sets is countable. This principle can be shown using a diagonal argument or by constructing a specific bijection. Consider that for each set \(A_i\) there is a bijection with \(\mathbb{N}\), i.e., each \(A_i\) can be listed as \(A_i = {a_{i1}, a_{i2}, a_{i3}, \ldots}\).
04
- Constructing a Bijection
Create a function that maps each element \(a_{ij}\) from the union set to a unique natural number. One common way is to list all elements in a table where rows correspond to the sets and columns to the elements within those sets, then traverse this table in a diagonal fashion (similar to Cantor's diagonal argument).
05
- Example of Diagonal Traversal
Illustrate the diagonal traversal process: \(a_{11}, a_{21}, a_{12}, a_{31}, a_{22}, a_{13}, a_{41}, \ldots\). Each element from the union will appear eventually because every set \(A_i\) is countable, and thus every combination \(a_{ij}\) will be reached in a finite number of steps.
06
- Conclusion
Since we can list all elements of the union \(A\) in a sequence, this shows a bijection between the union \(A\) and \(\mathbb{N}\). Hence, the union of countable sets is countable.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Cardinality
To understand countable sets, we first need to discuss **cardinality**. Cardinality refers to the size of a set, or more formally, the number of elements in the set. Sets can be finite, where the number of elements is a specific number, or infinite.
When we say a set has the same cardinality as the set of natural numbers, \(\backslashmathbb{N}\), we mean there exists a bijection (one-to-one correspondence) between the elements of the set and \(\backslashmathbb{N}\). For instance, the sets of natural numbers \(\backslashmathbb{N}\), integers \(\backslashmathbb{Z}\), and rational numbers \(\backslashmathbb{Q}\) all have the same cardinality, which is called countable infinity.
When we say a set has the same cardinality as the set of natural numbers, \(\backslashmathbb{N}\), we mean there exists a bijection (one-to-one correspondence) between the elements of the set and \(\backslashmathbb{N}\). For instance, the sets of natural numbers \(\backslashmathbb{N}\), integers \(\backslashmathbb{Z}\), and rational numbers \(\backslashmathbb{Q}\) all have the same cardinality, which is called countable infinity.
Union of Sets
The **union of sets** combines all elements from different sets into one larger set. If we have sets \(\backslash{A_1, A_2, A_3, \backslashldots\}\), the union is written as:
\[ A = \backslashbigcup_{i=1}^{\backslashinfty} A_i \]
This notation means that the set \A\ includes every element that is in any of the sets \A_i\. If each \A_i\ is countable, we want to determine if the union \A\ is also countable.
\[ A = \backslashbigcup_{i=1}^{\backslashinfty} A_i \]
This notation means that the set \A\ includes every element that is in any of the sets \A_i\. If each \A_i\ is countable, we want to determine if the union \A\ is also countable.
Bijection
A **bijection** is a function that pairs every element of one set exactly with one element of another set, and every element in the second set has a corresponding pair in the first set with no repetitions.
Constructing a bijection helps us prove that two sets have the same cardinality. For example, if we can map every element of a union set \A\ to a unique natural number, we show that \A\ is countable. In simpler terms, if we can list all elements of \A\ in an infinite sequence without missing any, we've established a bijection with \(\backslashmathbb{N}\).
Constructing a bijection helps us prove that two sets have the same cardinality. For example, if we can map every element of a union set \A\ to a unique natural number, we show that \A\ is countable. In simpler terms, if we can list all elements of \A\ in an infinite sequence without missing any, we've established a bijection with \(\backslashmathbb{N}\).
Diagonal Argument
The **diagonal argument** is a technique used to illustrate how to list elements from multiple countable sets in a specific order. Imagine we write the elements of each set \A_i\ in rows:
\[ \begin{align*} A_{1} &: a_{11}, a_{12}, a_{13}, \backslashldots \ \ A_{2} &: a_{21}, a_{22}, a_{23}, \backslashldots \ \ A_{3} &: a_{31}, a_{32}, a_{33}, \backslashldots \ \backslashldots \ \end{align*} \]
To cover all elements, we traverse this table diagonally: \a_{11}, a_{21}, a_{12}, a_{31}, a_{22}, a_{13}, \backslashldots\ By following this pattern, we ensure every element eventually appears in our sequence. Therefore, this method demonstrates that the union set is countable.
\[ \begin{align*} A_{1} &: a_{11}, a_{12}, a_{13}, \backslashldots \ \ A_{2} &: a_{21}, a_{22}, a_{23}, \backslashldots \ \ A_{3} &: a_{31}, a_{32}, a_{33}, \backslashldots \ \backslashldots \ \end{align*} \]
To cover all elements, we traverse this table diagonally: \a_{11}, a_{21}, a_{12}, a_{31}, a_{22}, a_{13}, \backslashldots\ By following this pattern, we ensure every element eventually appears in our sequence. Therefore, this method demonstrates that the union set is countable.
Countable Union Principle
The **countable union principle** asserts that the union of a countable number of countable sets is itself countable. This means if each set in the sequence \(\backslash{ A_i \backslash}_{i=1}^{\backslashinfty}\) is countable, their union is also countable.
We demonstrated this principle using a bijection and diagonal argument. Each set \A_i\ can be listed in a way that eventually includes every element in the union. Hence, by constructing a sequence that covers all elements of all sets, we show the final union set retains its countability.
We demonstrated this principle using a bijection and diagonal argument. Each set \A_i\ can be listed in a way that eventually includes every element in the union. Hence, by constructing a sequence that covers all elements of all sets, we show the final union set retains its countability.
