We are given the function $f: \mathbb{N} \to \mathbb{N}$, where $f(x) = 2x$, and we need to show that this function is one-to-one (injective) but not onto (surjective).

1. Proving the function is one-to-one (injective):

To prove that the function is injective, we need to show that if $f(x_1) = f(x_2)$, then $x_1 = x_2$. In other words, distinct inputs must produce distinct outputs.

• Assume $f(x_1) = f(x_2)$. This implies: $2x_1 = 2x_2$
• Divide both sides by 2: $x_1 = x_2$ Since $x_1 = x_2$, the function is injective, meaning it is one-to-one.

2. Proving the function is not onto (surjective):

To prove that the function is not surjective, we need to show that there exists at least one element in the codomain $\mathbb{N}$ (the set of natural numbers) that is not the image of any element in the domain $\mathbb{N}$.

• The function $f(x) = 2x$ maps every natural number $x$ to an even number $2x$. This means that all the outputs of the function are even natural numbers.
• However, the codomain of the function $f$ is all natural numbers $\mathbb{N}$, which includes both even and odd numbers.
• There is no $x \in \mathbb{N}$ such that $f(x)$ produces an odd number. For example, numbers like $1, 3, 5, \dots$ are in the codomain $\mathbb{N}$, but there is no $x \in \mathbb{N}$ such that $f(x) = 1$ or $f(x) = 3$.

Thus, the function is not surjective because it does not map to all elements of the codomain.

Conclusion:

• The function $f(x) = 2x$ is injective (one-to-one) but not surjective (onto).

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Would you like more details or have any further questions? Here are some related questions to deepen your understanding:

1. What are the conditions for a function to be both one-to-one and onto?
2. How would the function change if its codomain was restricted to only even natural numbers?
3. What is the difference between an injective and surjective function in terms of their properties?
4. How can we test if a function from $\mathbb{R} \to \mathbb{R}$ is injective and/or surjective?
5. How would you define the inverse of an injective function?

Tip: In general, to test injectivity, check if different inputs lead to different outputs. For surjectivity, verify if every element of the codomain is "hit" by some element in the domain.