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: 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.)
- 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}).
- 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}})