1. ## easy set theory

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2. Originally Posted by yellow4321
how could i show a bijection between the natural numbers and set of even natural numbers 2n?also from the N and Zintergers
Define $f:\cal{N} \to \cal{Z}$ 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

3. $\begin{array}{l}
\Phi :Z \mapsto N \\
\Phi (z) = \left\{ {\begin{array}{*{20}c}
{2\left| z \right|} & {z \le 0} \\
{2z - 1} & {z > 0} \\
\end{array}} \right. \\
\end{array}$

4. Originally Posted by yellow4321
how could i show a bijection between the natural numbers and set of even natural numbers 2n?
I'll start you off.

Let $f: \mathbb{N} \to \mathbb{N}_{\mbox{even}}$ be defined by $f(n) = 2n$ for all $n \in \mathbb{N}$.

We wish to show that $f$ is a bijective function. To show this we need to show:

(1) $f$ is one-to-one.

Definition: a function $f$ is one-to-one if and only if $f(a) = f(b)$ implies $a = b$

(2) $f$ is onto.

Definition: a function $f$ is onto if every element in the range is an image of some element in the domain, that is, $f: A \to B$ is onto if and only if for every $b \in B$ there is an $a \in A$ such that $b = f(a)$.

Now continue

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