This is an exam problem I gave once; you’ll need to know |A| is the size/number of elements of A. Let A and B be finite nonempty sets, and f a function from A to B. Fill one I, for injective (1-1), and one S, for surjective (onto), in each line of the following table to make the assertions correct.

If	f must be	f may or may not be	f cannot be
a) |A| < |B|
b) |A| > |B|
c) |A| = 1
d) |B| = 1

e) Working only from the definitions of function, surjectivity, and injectivity, and not from other prior results, prove your answer for either line c) or line d) above. If there is an entry in the middle column, pin down the conditions under which that condition will hold of f.

No answers on this one, but a hint for part e): think of the function as a relation. That’s often helpful in working with functions between finite sets.