Question

Problem 1. Give an example of a function f:Z→N which is onto but not one-to-one (and prove that f has these desired properties).

Answer 1

Main Answer:

To provide an example of a function
`\(f: \mathbb{Z} \rightarrow \mathbb{N}\)` that is onto but not one-to-one, consider the function
`\(f(x) = |x|\)`.

Step-by-step explanation:

The function
`\(f: \mathbb{Z} \rightarrow \mathbb{N}\)` is defined as follows:

`\[ f(x) = \begin{cases} x, & \text{if } x \geq 0 \\-x, & \text{if } x < 0 \end{cases}\]`

Onto:

For any
`\(y\)` in the codomain
`\(\mathbb{N}\)`, there exists at least one
`\(x\)` in the domain
`\(\mathbb{Z}\)` such that
`\(f(x) = y\)`. For instance, for
`\(y = 5\)`,
`\(f(5) = 5\)`, and for
`\(y = 3\), \(f(-3) = 3\)`, showing that
`\(f\)` is onto.

Not One-to-One:

The function is not one-to-one because multiple elements in the domain can map to the same element in the codomain. For example, both
`\(f(2) = 2\)` and
`\(f(-2) = 2\)`, violating the one-to-one property.

Thus, the function
`\(f(x) = |x|\)` serves as an example of a function that is onto but not one-to-one from
`\(\mathbb{Z}\)` to
`\(\mathbb{N}\)`.