Show that the function `f: N->N` given by, `f(n)=n-(-1)^n` for all `n in N` is a bijection.

Explanation

To show that the function \( f: \mathbb{N} \to \mathbb{N} \) defined by \( f(n) = n - (-1)^n \) is a bijection, we need to demonstrate that it is both injective (one-to-one) and surjective (onto).

### Step 1: Determine the function's behavior for odd and even \( n \)

1. **For odd \( n \)**:
- If \( n \) is odd, \( (-1)^n = -1 \).
- Therefore, \( f(n) = n - (-1) = n + 1 \).
- This means odd numbers map to the next even number.
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