1. f1 and f4 are functions. f2 and f3 are not.

2. Letting A and B be the left and right sets. \\(\forall a \in A, f1(a) \in B \wedge f4(a) \in B \\). However, for both f3 and f2 \\(\exists a \in A \text{ s.t. } f(a) \notin B\\). For f2 it's \\(\circ\\), for f3 it's \\(\bullet\\).

3. To have an injective, but not surjective, function \\(|A| \lt |B|\\). An example of this would be \\(A = \\{1,2,3\\}\\) and \\(B = \\{1,2,3,4,5,6\\}\\) with \\(f (a) = 2a\\). for each b in B either there is a (singular) a in A which maps to it or no a maps to it (at most one).

4. To have a surjective, but not injective, function we require the opposite: \\(|A| \gt |B|\\). Using the same two sets from part 3, we can define \\(g(b) = \lceil \frac{b}{2} \rceil\\). Each of the 3 elements of A will be mapped to, but each by 2 elements of B.

If the sizes of A and B are equal, then we can find functions which are both surjective and injective or functions which are neither. However we cannot find functions which are one or the other.

2. Letting A and B be the left and right sets. \\(\forall a \in A, f1(a) \in B \wedge f4(a) \in B \\). However, for both f3 and f2 \\(\exists a \in A \text{ s.t. } f(a) \notin B\\). For f2 it's \\(\circ\\), for f3 it's \\(\bullet\\).

3. To have an injective, but not surjective, function \\(|A| \lt |B|\\). An example of this would be \\(A = \\{1,2,3\\}\\) and \\(B = \\{1,2,3,4,5,6\\}\\) with \\(f (a) = 2a\\). for each b in B either there is a (singular) a in A which maps to it or no a maps to it (at most one).

4. To have a surjective, but not injective, function we require the opposite: \\(|A| \gt |B|\\). Using the same two sets from part 3, we can define \\(g(b) = \lceil \frac{b}{2} \rceil\\). Each of the 3 elements of A will be mapped to, but each by 2 elements of B.

If the sizes of A and B are equal, then we can find functions which are both surjective and injective or functions which are neither. However we cannot find functions which are one or the other.