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This correct graphical representation of this proof?

The graphical representation you provided is not correct for proving the injectivity of the function \( f(n) = (-1)^n \cdot n \). 

Let's analyze why this is the case and what a correct graphical representation would look like.

### Analyzing the Given Graphs
1. **Top Graph:**
   - This graph shows elements from the domain \( n_1 \) mapping to elements in the codomain labeled \( f(n) \).
   - However, it inaccurately shows multiple domain elements mapping to the same codomain element, which implies non-injectivity.

2. **Bottom Graph:**
   - This graph is similar to the top one but does not correctly reflect the injective nature of the function \( f(n) = (-1)^n \cdot n \).

### Correct Graphical Representation
To correctly demonstrate that the function \( f(n) = (-1)^n \cdot n \) is injective, each unique element in the domain \(\mathbb{Z}^+\) should map to a unique element in the codomain \(\mathbb{Z}\). Here’s how the correct representation would look like:

1. **Domain (\(\mathbb{Z}^+\)):** Positive integers \( \{1, 2, 3, 4, 5, \ldots \} \)
2. **Codomain (\(\mathbb{Z}\)):** Integers \(\{\ldots, -4, -3, -2, -1, 0, 1, 2, 3, 4, \ldots \}\)

### Steps for Correct Representation
1. **Map each \( n \) to \( (-1)^n \cdot n \):**
   - \( f(1) = (-1)^1 \cdot 1 = -1 \)
   - \( f(2) = (-1)^2 \cdot 2 = 2 \)
   - \( f(3) = (-1)^3 \cdot 3 = -3 \)
   - \( f(4) = (-1)^4 \cdot 4 = 4 \)
   - And so on.

2. **Verify injectivity:**
   - Each \( n \) in the domain should map to a unique value in the codomain.
   - No two different \( n \) values should map to the same codomain value.

### Correct Graphical Diagram
- **Elements in the domain**: \( \{1, 2, 3, 4, 5, \ldots\} \)
- **Mappings**:
  - \( 1 \rightarrow -1 \)
  - \( 2 \rightarrow 2 \)
  - \( 3 \rightarrow -3 \)
  - \( 4 \rightarrow 4 \)
  - \( 5 \rightarrow -5 \)

This ensures that each input \( n \) maps to a unique output in the codomain, proving that the function is injective.

### Diagram

Here is a simple graphical representation:

```
Domain (Z^+):    Codomain (Z):
 1 ---------->   -1
 2 ---------->    2
 3 ---------->   -3
 4 ---------->    4
 5 ---------->   -5
...              ...
```

This representation clearly shows that each element from the domain maps to a unique element in the codomain, thus proving injectivity.

Would you like more details or have any other questions? Here are some questions you might want to ask next:

1. How do you prove a function is surjective?
2. What is the difference between injective, surjective, and bijective functions?
3. Can you explain the concept of mathematical induction?
4. How do you find the inverse of a function?
5. What are the properties of even and odd functions?

**Tip:** Visual aids, such as mappings or diagrams, can be very helpful in understanding and proving the properties of functions.

Understand how to correctly represent the function f(n) = (-1)^n * n graphically to prove its injectivity. Learn the steps and diagrams needed to demonstrate that each element in the domain maps uniquely to the codomain, essential for mathematical proofs.

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