.
.
Define as
..........n/2 ----------when n is even
f(n) =
..........(1-n)/2 ---------when n is odd
The first one is a simple exercise to show injectivity and surjectivity. Do it for this too.
P.S: How do we put the huge bracket for such piecewise defined function in TeX? Thanks
I'll start you off.
Let be defined by for all .
We wish to show that is a bijective function. To show this we need to show:
(1) is one-to-one.
Definition: a function is one-to-one if and only if implies
(2) is onto.
Definition: a function is onto if every element in the range is an image of some element in the domain, that is, is onto if and only if for every there is an such that .
Now continue