1. ## bijective maps

Let $n \in \mathbb{Z^+}, X$ is nonempty.
(i) Find a bijective map $f: X^\omega \times X^\omega \rightarrow X^\omega$
(ii) If $A \subset B$ find an injective map $gA^\omega)^n \rightarrow B^\omega" alt="gA^\omega)^n \rightarrow B^\omega" />

The cartesian product $A_1 \times A_2 \times ...$ is the set of all $\omega-$tuples of elements of $X$ is denoted by $X^\omega$.

I've tried many examples but none works... Can I get some help with these two please?

2. (i) define $f$ for example like this: $f(\left)=u$ where $u(n) = \begin{cases} g(m) & \mbox{if } n=2m \\ h(m) & \mbox{if } n=2m+1 \end{cases}$

(ii) if $A$ is empty then $g = \emptyset$ (the empty mapping) is injective. if $A$ is nonempty, you can define g similarly as in (i):

$g(\left) = v$ where $v(i) = \begin{cases} h_0(m) & \mbox{if } i=nm \\ h_1(m) & \mbox{if } i=nm+1 \\ \cdots \\ h_{n-1}(m) & \mbox{if } i=nm+n-1\end{cases}$

3. Originally Posted by dori1123
Let $n \in \mathbb{Z^+}, X$ is nonempty.
(i) Find a bijective map $f: X^\omega \times X^\omega \rightarrow X^\omega$
This makes no sense. Did you mean $X\subset Z^+$ rather than $n \in Z^+$?

(ii) If $A \subset B$ find an injective map $gA^\omega)^n \rightarrow B^\omega" alt="gA^\omega)^n \rightarrow B^\omega" />

The cartesian product $A_1 \times A_2 \times ...$ is the set of all $\omega-$tuples of elements of $X$ is denoted by $X^\omega$.

I've tried many examples but none works... Can I get some help with these two please?

4. Originally Posted by HallsofIvy
This makes no sense. Did you mean $X\subset Z^+$ rather than $n \in Z^+$?
Hi,
what exactly doesn't make sense? condition $n \in \mathbb{Z}^+$ applies to part (ii).

5. it is $n \in \mathbb{Z^+}$, it's a problem from the book Topology by Munkres

6. When you write $f(\left) = ...$, is this $\left$ the cyclic group generated by $g$ and $h$?

7. no, it is an ordered pair.
$f: X^\omega \times X^\omega \rightarrow X^\omega$ means that $f$ takes a pair of functions from $\omega$ to $X$ and returns another function from $\omega$ to $X$.

8. Thanks I just want to make sure because I've been using the symbol < > for cyclic groups..

Can you also help me with one more problem?
with the same conditions given,
find a bijective map $f: X^n \times X^\omega \rightarrow X^\omega$

9. if you understand previous two solutions, this one should be no problem for you..
define $f(\left)=u$ where $u(i) = \begin{cases} g(i) & \mbox{if } i

can you verify it is well-defined, injective and surjective?