Show that the whole numbers z is counterable

To show that the set of whole numbers \( \mathbb{Z} \) is countable, we need to demonstrate that there exists a one-to-one correspondence (a bijection) between the set of whole numbers \( \mathbb{Z} \) and the set of natural numbers \( \mathbb{N} \) (which includes 0, i.e., \( \mathbb{N} = \{0, 1, 2, 3, \dots\} \)).

### Steps to Prove \( \mathbb{Z} \) is Countable:

1. **List the Whole Numbers \( \mathbb{Z} \)**:
   The set of whole numbers \( \mathbb{Z} \) includes all integers, both positive and negative, and zero:
   \[
   \mathbb{Z} = \{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}
   \]

2. **Interleave Positive and Negative Numbers**:
   We can list the elements of \( \mathbb{Z} \) in a specific order that allows us to create a one-to-one correspondence with \( \mathbb{N} \). One possible order is:
   \[
   0, 1, -1, 2, -2, 3, -3, 4, -4, \dots
   \]
   This sequence interleaves positive and negative integers starting with zero.

3. **Define a Bijection \( f: \mathbb{N} \to \mathbb{Z} \)**:
   Define a function \( f \) that maps each natural number \( n \in \mathbb{N} \) to an integer in \( \mathbb{Z} \) according to the pattern observed above:
   \[
   f(n) = 
   \begin{cases} 
   \frac{n}{2} & \text{if } n \text{ is even} \\
   -\frac{n+1}{2} & \text{if } n \text{ is odd}
   \end{cases}
   \]
   - For \( n = 0 \), \( f(0) = 0 \).
   - For \( n = 1 \), \( f(1) = -1 \).
   - For \( n = 2 \), \( f(2) = 1 \).
   - For \( n = 3 \), \( f(3) = -2 \), and so on.

4. **Show that \( f \) is a Bijection**:
   - **Injective (One-to-One)**: If \( f(n_1) = f(n_2) \), then \( n_1 = n_2 \). This means no two different natural numbers map to the same integer.
   - **Surjective (Onto)**: For every integer \( z \in \mathbb{Z} \), there exists a natural number \( n \in \mathbb{N} \) such that \( f(n) = z \). This ensures that every integer is covered by some natural number.

Since the function \( f \) is both injective and surjective, it is a bijection.

### Conclusion:
The set \( \mathbb{Z} \) of whole numbers is countable because there exists a bijection between \( \mathbb{N} \) and \( \mathbb{Z} \). Hence, \( \mathbb{Z} \) is countable.

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Would you like more details or have any questions? 

Here are some related questions for you:

1. How do you prove that the set of rational numbers \( \mathbb{Q} \) is countable?
2. Can you explain the concept of uncountable sets with an example?
3. How is the Cantor diagonalization method used to show that real numbers are uncountable?
4. What is the difference between countable and uncountable infinity?
5. Can the set of all finite subsets of \( \mathbb{Z} \) be countable?

**Tip**: A set is countable if you can list its elements in a sequence, even if the sequence is infinite.

Learn how to prove that the set of whole numbers (integers) is countable. Understand the concept of bijection and how it establishes a one-to-one correspondence between the set of natural numbers and the set of whole numbers. Suitable for advanced high school students and anyone interested in set theory and mathematical proofs.

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