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.
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- [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.