Let's analyze each of the given functions to determine whether they are onto.

A function $f: A \to B$ is onto (or surjective) if for every element $b \in B$, there is at least one element $a \in A$ such that $f(a) = b$. In this case, we are dealing with the domain $Z \times Z$ (pairs of integers) and the codomain $Z$ (integers).

(a) $f(m, n) = m + n$

For any $z \in Z$, we want to find integers $m$ and $n$ such that $m + n = z$.

• This is possible for every integer $z$, since we can always choose, for example, $m = z$ and $n = 0$, or $m = 0$ and $n = z$.

Thus, the function is onto.

(b) $f(m, n) = m^2 + n^2$

We want to check if every integer $z \in Z$ can be written as $m^2 + n^2$ for some integers $m$ and $n$.

• Notice that $m^2 + n^2 \geq 0$ for all integers $m$ and $n$, meaning that the function can never output a negative integer.

Therefore, this function is not onto because negative integers cannot be obtained as values of $f(m, n)$.

(c) $f(m, n) = m$

For any integer $z \in Z$, we want to find a pair $(m, n)$ such that $f(m, n) = m = z$.

• We can always choose $m = z$ and any value for $n$, so every integer $z \in Z$ can be reached.

Thus, the function is onto.

(d) $f(m, n) = |n|$

We need to check if for every integer $z \in Z$, there are integers $m$ and $n$ such that $|n| = z$.

• $|n|$ is always non-negative, meaning that $f(m, n)$ can never output a negative integer. Thus, negative integers are not in the image of this function.

Therefore, this function is not onto because negative integers cannot be reached.

(e) $f(m, n) = m - n$

We want to find integers $m$ and $n$ such that $m - n = z$ for any integer $z$.

• This is possible for any integer $z$, as we can simply choose $m = z + n$, which will yield $m - n = z$ for any $n$.

Thus, the function is onto.

Summary of Results:

• (a) $f(m, n) = m + n$ is onto.
• (b) $f(m, n) = m^2 + n^2$ is not onto.
• (c) $f(m, n) = m$ is onto.
• (d) $f(m, n) = |n|$ is not onto.
• (e) $f(m, n) = m - n$ is onto.

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Would you like any further clarification on these results, or additional questions on related topics?

Here are some related questions you might find interesting:

1. What is the difference between injective and surjective functions?
2. How can we prove that a function is not onto?
3. What are the properties of a bijection (both injective and surjective)?
4. How does the concept of onto functions apply to other sets, such as real numbers or complex numbers?
5. Can a function defined on a finite set be onto if the codomain is infinite?

Tip: When testing if a function is onto, it's helpful to think about the range of values that the function can take. If there are values in the codomain that cannot be reached by any input, the function is not onto.