# Math 417 Homework 1: Due Friday January 24

*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] If X and Y are sets and f : X -> Y and g : Y -> X
are functions such that fg = 1
_{Y} and gf = 1_{X},
show that f is bijective.
- [2] Let X be a nonempty set. If f and g are elements of
Perm(X), show that fg is too. (Hint: First show fg is injective,
then show fg is surjective. You can cite results from the book,
if that is helpful, but then be sure you understand the book's
proof of the result you cite!)
- [3] If X is a finite set of |X| = n elements, show
that |Perm(X)| = n!.
- [4] Give two reasons why the set of all odd integers
does not form a group under the operation of addition.
- [5] Let X be the set of all real numbers except -1 (i.e.,
X =
**R** - {-1}). Define a binary operation * on X by the rule
a*b = ab + a + b. (Convince yourself that * is a binary operation!
The only problem is if a*b = -1, but in fact that never happens.)
- (a) Does * have an identity element? If so, what is it?
- (b) Is * associative? Why or why not?
- Is X a group under *? Why or why not?