Exercise 1) Define two of your favorite sets (numbers, household objects, children, whatever), and define some a) injective functions between them (make sure to specify where the function goes from and where it goes to) b) surjective functions between them, and c) bijective functions between them. Is it always possible to do this for each case? If not, why not?
Solution: This is another exercise that does not really have a “right” solution, so I’ll just give an example that I can think of right now to get the ball rolling. I’m sitting in a coffee shop as I write this and it’s raining outside, so let me define A to be the set of all rain drops that fall within 5 feet of the window to the right of me in the next 10 seconds, and let me define B to be the set of paintings on the wall. Seeing as it’s raining pretty hard, I’m going to assume that the set A has many more elements than the set B. I therefore won’t be able to define an injective function from A to B, but I can do so from B to A. Let me define a function from B to A by sending the weird abstract portrait in front of me to the first rain drop that ended up hitting the sidewalk, and let me send the landscape of the strange shadowy trees to the 5th rain drop to hit the sidewalk. Finally, the huge painting behind me consisting of one large yellow stripe and one slightly darker yellow strip will be sent to the 27th rain drop to hit the sidewalk. This is then injective since I never sent any two distinct paintings to the same rain drop, but it is far from surjective (and therefore far from bijective). Seeing as there are infinitely many such examples, I’ll just stop here.