Final answer:
The functions (a) f (m, n) = 2m - n and (b) f (m, n) = m^2 - n^2 are onto since for any integer z, we can find values of m and n that satisfy the equation. Functions (c) f (m, n) = mn + 1, (d) f (m, n) = |m| - |n|, and (e) f (m, n) = m^2 - 4 are not onto because they cannot produce all integer values as outputs.
Step-by-step explanation:
To determine whether f : z × z → z is onto for the given functions, we examine each function:
- (a) f (m, n) = 2m - n: This function is onto because for any integer value z, you can choose m and n such that 2m - n = z. Set m = z and n = 0, then f(m, n) = 2z, which can be any integer.
- (b) f (m, n) = m^2 - n^2: This function is also onto. For any integer z, there exist integers m and n that will make m^2 - n^2 = z. For example, to get z as 0, let m = n.
- (c) f (m, n) = mn + 1: This function is not onto, since when m and n are both integers, mn will be an integer, and adding 1 will never yield an even integer.
- (d) f (m, n) = |m| - |n|: This function is not onto since the absolute values ensure the result is always nonnegative, and so negative integers aren't in the range.
- (e) f (m, n) = m^2 - 4: This function is not onto because it will always generate outcomes that are either 0 or negative when squared, and subtracting 4 will not cover all integers.
Option 2) f is onto for some values of m and n applies to (a) and (b). Option 3) f is not onto for any values of m and n applies to (c), (d), and (e).