give me two different bijective functions from n to z

Sure, here are two different bijective functions from \(\mathbb{N}\) (the set of natural numbers) to \(\mathbb{Z}\) (the set of integers).

### Function 1
\[ f_1(n) = 
\begin{cases} 
\frac{n}{2} & \text{if } n \text{ is even} \\
-\frac{n+1}{2} & \text{if } n \text{ is odd}
\end{cases}
\]

**Explanation:**
- For even \( n \), the function maps to positive integers.
- For odd \( n \), the function maps to negative integers.

### Function 2
\[ f_2(n) = 
\begin{cases} 
\frac{n-1}{2} & \text{if } n \text{ is odd} \\
-\frac{n}{2} & \text{if } n \text{ is even}
\end{cases}
\]

**Explanation:**
- For odd \( n \), the function maps to non-negative integers.
- For even \( n \), the function maps to negative integers.

Both functions ensure that each natural number \( n \) maps to a unique integer \( z \), covering all integers without repetition or omission, making them bijective.

Would you like details on how these functions work, or do you have any specific questions?

Here are some questions you might want to explore next:
1. How can we prove that the given functions are bijective?
2. What are some other examples of bijective functions between different sets?
3. How can we visualize these bijective functions on a number line?
4. What are the properties of bijective functions?
5. Can we define a bijective function from \(\mathbb{R}\) to \(\mathbb{R}\)?

**Tip:** When defining bijective functions, it's crucial to ensure that every element in the target set is mapped by exactly one element in the domain and vice versa. This guarantees both injectivity (one-to-one) and surjectivity (onto).

Explore two different bijective functions from natural numbers to integers (whole numbers), ensuring each natural number maps uniquely to an integer. Understand how these functions work and their significance in number theory.

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