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Define $\displaystyle f:\cal{N} \to \cal{Z}$ 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 $\displaystyle f: \mathbb{N} \to \mathbb{N}_{\mbox{even}}$ be defined by $\displaystyle f(n) = 2n$ for all $\displaystyle n \in \mathbb{N}$.
We wish to show that $\displaystyle f$ is a bijective function. To show this we need to show:
(1) $\displaystyle f$ is one-to-one.
Definition: a function $\displaystyle f$ is one-to-one if and only if $\displaystyle f(a) = f(b)$ implies $\displaystyle a = b$
(2) $\displaystyle f$ is onto.
Definition: a function $\displaystyle f$ is onto if every element in the range is an image of some element in the domain, that is, $\displaystyle f: A \to B$ is onto if and only if for every $\displaystyle b \in B$ there is an $\displaystyle a \in A$ such that $\displaystyle b = f(a)$.
Now continue