show subsets of natural numbers are countable

To show that any subset of natural numbers is countable, we can break down the explanation step by step:

### 1. **Definition of Countability**
A set is **countable** if there exists a one-to-one correspondence (bijection) between the elements of the set and the set of natural numbers \( \mathbb{N} \) (i.e., \( \{1, 2, 3, \dots\} \)). This means we can "list" the elements of the set in an ordered sequence \( a_1, a_2, a_3, \dots \), where each element corresponds to a unique natural number.

There are two main categories of countable sets:
- **Finite sets**: Any finite set is trivially countable because we can assign each element a unique natural number.
- **Infinite countable sets**: These are infinite sets that still have a bijection with \( \mathbb{N} \).

### 2. **Natural Numbers \( \mathbb{N} \) are Countable**
By definition, \( \mathbb{N} \) is countable because every natural number can be listed in a sequence \( 1, 2, 3, 4, \dots \), and each number corresponds to itself.

### 3. **Subsets of Natural Numbers**
Consider any subset \( A \) of natural numbers, which can either be:
- **Finite**: A finite subset of \( \mathbb{N} \) (e.g., \( A = \{1, 3, 5\} \)) is trivially countable because there is a finite number of elements, and we can always map each element to the first few natural numbers.
  
- **Infinite**: If \( A \) is an infinite subset of \( \mathbb{N} \), it is still countable. Why? Because for any infinite subset \( A \subseteq \mathbb{N} \), we can still list its elements in an increasing sequence \( a_1, a_2, a_3, \dots \), where \( a_i \in A \) for each \( i \), and thus create a bijection with \( \mathbb{N} \).

#### Example of an Infinite Subset
Let \( A = \{2, 4, 6, 8, \dots\} \), the set of even natural numbers. We can define a function:
\[
f(n) = 2n
\]
This function takes every natural number \( n \) and maps it to an even number \( 2n \). Since this is a one-to-one correspondence (bijection) between \( \mathbb{N} \) and the set of even numbers, \( A \) is countable.

### 4. **Conclusion**
Any subset of natural numbers, whether finite or infinite, is countable. For finite sets, the countability is trivial. For infinite subsets, we can always list the elements in some ordered sequence that corresponds to the natural numbers.

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### 5 Related Questions:
1. What is the difference between countable and uncountable sets?
2. Can the set of real numbers \( \mathbb{R} \) be considered countable?
3. How can we prove that rational numbers are countable?
4. Are all infinite sets countable?
5. What is the diagonalization argument for proving uncountability?

**Tip**: When checking if a set is countable, look for a way to list all elements of the set in a sequence.

Learn how to prove that any subset of natural numbers is countable with a step-by-step breakdown. This explanation covers key concepts in set theory, such as bijections and countability, with examples of both finite and infinite subsets. Suitable for advanced high school students (Grades 11-12).

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