One To One Not Onto

1. **Problem Statement:** Define a function $f: \mathbb{N} \to \mathbb{N}$ that is one-to-one but not onto. 2. **Formula and Explanation:** Consider the function $$f(n) = 2n$$ where $n \in \mathbb{N}$. 3. **One-to-One (Injective) Check:** A function is one-to-one if for any $n_1, n_2 \in \mathbb{N}$, $$f(n_1) = f(n_2) \implies n_1 = n_2$$ Here, $$f(n_1) = 2n_1, \quad f(n_2) = 2n_2$$ If $2n_1 = 2n_2$, dividing both sides by 2 gives $$n_1 = n_2$$ Thus, $f$ is injective. 4. **Not Onto (Not Surjective) Check:** A function is onto if for every $m \in \mathbb{N}$, there exists $n \in \mathbb{N}$ such that $$f(n) = m$$ Here, $f(n) = 2n$ only produces even numbers. Odd numbers in $\mathbb{N}$ have no pre-image. Therefore, $f$ is not onto. 5. **Conclusion:** The function $f(n) = 2n$ is one-to-one but not onto.