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 $\displaystyle f:\cal{N} \to \cal{Z}$ 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

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

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 $\displaystyle f: \mathbb{N} \to \mathbb{N}_{\mbox{even}}$ 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

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