Math 417 Homework 3: Due Friday February 7

Instructions: You can discuss these problems with others, but write up your solutions on your own (i.e., don't just copy someone else's solutions, else the feedback I give you won't help you much). Please be neat and write in full sentences.

• [1] For this problem, let h: A ® B be a function from a set A to a set B. [Note: the symbol ® should be a function arrow.] We will need some definitions.

Definition 1: For any subset C of A, define h(C) to be the subset { h(a) : a is an element of C } of B; we call h(C) the image of C under h. It is just the set of all values of h that you get by plugging in elements from C. (Thus given a function h from A to B, we also get a function from the set Subsets(A) of subsets of A to the set Subsets(B) of subsets of B, which is also usually denoted h : Subsets(A) ® Subsets(B). Using the same name for two different things is usually verboten because it is too confusing, but this is what's done in this case. It is not usually a problem, because you can tell which h is meant by seeing what is being plugged into it, either an element or a subset.)

Definition 2: For any subset D of B, define h-1(D) to be the subset { a in A : h(a) is in D } of A, called the inverse image of D under h. Note that this defines a function h-1 : Subsets(B) ® Subsets(A) from the set of subsets of B to the set of subsets of A. (The notation h-1 for the inverse image function is unfortunate, since the same notation is used for the inverse function, when it exists: not all functions have an inverse under composition, only the bijective ones do. However, for any function you always have the inverse image function. Again you can tell whether h-1 refers to the inverse function or the inverse image function by whether what is being plugged in is an element of B or a subset of B.)

• (a) If f: R ® R is the function f(x) = x2, find f([1,4]) and f -1([1,4]), where [a,b] denotes the interval a <= x <= b.

• (b) If h: A ® B is surjective, show that h-1 : Subsets(B) ® Subsets(A) is injective.

• (c) If h: A ® B is injective, is it true that h-1 : Subsets(B) ® Subsets(A) is surjective? Prove it if it is true, or give a counterexample if it is false.

• (d) Given any subsets E and F of A, show that h(EÇF) is a subset of h(E)Çh(F). Give an example to show that h(EÇF) = h(E)Çh(F) can be false. [Note: Ç should be an intersection symbol.]
• [2] Translate each of the following expressions into additive notation: (a) a-2(b-1c)2; (b) (ab2)-3c2 = e.

• [3] Let g be an element of a group G. Define functions
lg : G ® G and rg : G ® G by left and right multiplication, respectively; i.e., lg(x) = gx and rg(x) = xg. Show that lg and rg are bijective. [Note: l should be a Greek letter lambda, and r should be a Greek letter rho.] Use the bijectivity to fill in the following group multiplication table. From the filled in table, determine whether or not the group is abelian:

 e a b c d e e a b e b c d e c d a b d

• [4] Let G be a group such that if a, b and c belong to G, and if ab = ca, then b must equal c. Show that G is abelian.