Question

Determine whether the function f : z × z → z is onto if

a.f (m, n) = m + n.
b.f (m, n) = m2 + n2.
c.f (m, n) = m.
d.f (m, n) = |n|.
e.f (m, n) = m − n

Answer 1

The functions f(m, n) = m + n, f(m, n) = m, and f(m, n) = m - n are onto since for any integer z, there exists a pair (m, n) that maps to z. However, f(m, n) = m^2 + n^2 and f(m, n) = |n| are not onto as negative integers have no preimage under these functions.

Step-by-step explanation:

When analyzing the function f : ℤ × ℤ → ℤ, we are looking at a function from the set of integers times the set of integers to the set of integers. We aim to determine if these functions are on.

• To test if f(m, n) = m + n is onto, for any integer z, we can set m = z and n = 0, which would give us f(m, n) = z. Hence, this function is on.
• For the function f(m, n) = m^2 + n^2, we cannot get negative integers as outputs, which are in an integer set; therefore, this function is not onto.
• In the case of f(m, n) = m, for every integer z, we can set m = z, making this function onto.
• The function f(m, n) = |n| cannot give negative integers, which means it's not onto.
• Lastly, for f(m, n) = m - n, any integer z can be expressed by setting m = z + n. Therefore, this function is onto.

Answer 2

Assuming "z" is
`\mathbb Z`, the set of integers:

a. yes
b. no;
`m^2+n^2` will always be a non-negative integer
c. yes
d. no; same reason as (b)
e. yes