HW #2 due Wednesday April 30 (acceptable Friday)

Let f:X->Y and g:Y->Z be functions.

1. Give an example where g o f is onto, but f is not onto.

2. Assume g o f is 1:1, and f is onto. Show that g is 1:1.

3. Let X have 2 elements and Y have n elements. Show, by induction on n, that the number of 1:1 functions from X to Y is n²-n.

4. Assume that X=Z, so g:Y->X.
The function I:X->X such that I(x)=x for all x in X is called the identity function.
(Every set's got one!)
Call g a left inverse of f if g o f is the identity function on X.

a. If g is a left inverse of f, show that f is 1:1.
b. If f is 1:1 and X is nonempty, show that f has a left inverse.
c. If X is empty, show that f is 1:1, and determine when it has a left inverse.