1) Define a function f : S13 S13 by f(a) = 6 mod 13. a) What is the domain of f? The range of ? The image of f? b) Prove that f is injective, using the formal definition of injectivity, c) Is f invertible? If so, give the values of its inverse function. 2) Define a function g : Sis H Sis by g(x) = fx mod 15. a) What is the domain of g? The range of g? The image of g? b) Show that g is not injective. If x'i and x2 in Sis map to the same value under g, what must be true about 2 and 2? c) Show that g is not surjective. If y e sis is not mapped to by g, what must be true of y? d) Give a set AC Sis and BC S15, such that g is an invertible map from A to B. Find the largest A, B you can. Given a function / : S - S (unrelated to the above), a value xe S is a fixed point or has order 1 if x = S(x). Similarly, a has onler 2 if x = f(f(x)), and order 3 if x = ( ())). A point & has order k if applying f k-times to x yields x. If never returns to x under repeated applications of f, then x has infinite order, or is transitory. 3) For any N > 1, give an example of an f : Sv H Sy that has no fixed points. 4) For any N > 1, give an example of an f: Sy + Sy that has only a single point of finite order. 5) Argue that for any N > 1, given f :S H Sw, if x has finite order, then x has an inverse under f, i.e., if every pointxe Sy has a finite order, then f is invertible. Consider small eramples first