## Extra Credit

If you missed any points on problems 2 or 3 of Homework 5, here's a chance to get some or all of them back. If you lost x points on problems 2 and 3 of Homework 5 and you get a score of y% on this extra credit assignment, I'll add xy/100 (rounded up to the nearest integer) to your Homework 5 score.
1. Consider the function f: Z6 -> Z6 defined by f(x) = x2 - x mod 6. Find f -1({0}). I.e., find all x such that f(x) = 0. (You should find that f(x) = 0 has four roots, which is interesting, since in normal arithmetic a quadratic equation has at most 2 roots.)
2. Now consider the function g: Z6 -> Z6 defined by g(x) = x2 mod 6.
• (a) Find g({1,2,3,4}).
• (b) Find g-1({2,5}).
3. Given any function h: A -> B, we get a function h-1 : 2B -> 2A. But for any function, you get the associated inverse image function, so h-1 gives yet another function (h-1)-1 : 22A -> 22B. Determine which of the following make sense, and evaluate those that do, where g is as above.
• (a) g-1({0})
• (b) g-1(0)
• (c) (g-1)-1({0})
• (d) (g-1)-1({{0}})