https://file.aiquickdraw.com/mathbot/file/1721919805930a27t742h.jpg

I drew both cases 1st one is if both odd or even and 2nd one is if one is odd and another is even and vice-versa. Is this correct representation according to solution?

Yes, your representation is correct according to the solution. Let's break down your diagrams for clarity:

### Diagram Analysis

#### First Diagram (Top)
This diagram shows the case where \( n_1 \) and \( n_2 \) are both odd or both even:

- The left set (a) represents possible values for \( n \), which can be 1 or -1.
- The right set (b) also represents the same possible values for \( f(n) \), which can be 1 or -1.
- If \( n_1 \) and \( n_2 \) are both even or both odd, \( f(n_1) \) and \( f(n_2) \) map to the same point in \( \mathbb{Z} \), ensuring \( n_1 = n_2 \).

#### Second Diagram (Bottom)
This diagram shows the case where one of \( n_1 \) or \( n_2 \) is odd and the other is even:

- The left set (a) represents possible values for \( n \), which can be 1 or -1.
- The right set (b) also represents the same possible values for \( f(n) \), which can be 1 or -1.
- If \( n_1 \) is odd and \( n_2 \) is even, \( f(n_1) \) maps to one point (say, 1) and \( f(n_2) \) maps to another point (say, -1), ensuring that \( n_1 \neq n_2 \).

Your drawings correctly illustrate the cases discussed in the solution:

1. When \( n_1 \) and \( n_2 \) are both odd or both even, they map to the same point.
2. When \( n_1 \) is odd and \( n_2 \) is even (or vice versa), they map to different points.

Therefore, your visual representation aligns with the proof and accurately depicts the injectivity of the function \( f(n) = (-1)^n \cdot n \).

Do you want details or have any questions? Here are some more questions you might find interesting:

1. How do you prove that a function is surjective?
2. What is the difference between injective, surjective, and bijective functions?
3. How can you determine if a function is bijective?
4. What are some examples of non-injective functions?
5. How do you prove the injectivity of functions involving trigonometric identities?

**Tip:** Visual aids like diagrams can be very helpful in understanding and proving properties of functions, especially injectivity and surjectivity.

Explore the concept of function injectivity through visual analysis and examples. Learn how to visually represent cases where functions are injective based on whether inputs are odd or even. Detailed diagrams illustrate the scenarios and proofs. Ideal for advanced mathematics enthusiasts.

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