## 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.
- Consider the function f:
**Z**_{6} -> **Z**_{6}
defined by f(x) = x^{2} - 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.)
- Now consider the function g:
**Z**_{6} -> **Z**_{6}
defined by g(x) = x^{2} mod 6.
- (a) Find g({1,2,3,4}).
- (b) Find g
^{-1}({2,5}).

- Given any function h: A -> B, we get a function
h
^{-1} : 2^{B} -> 2^{A}. But for any function,
you get the associated inverse image function, so h^{-1}
gives yet another function (h^{-1})^{-1} : 2^{2A} -> 2^{2B}.
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}})