How to prove show that $\mathbb{R}\setminus \{0\}$ is isomorphic to $ \mathbb{R}\times \mathbb{Z}_2$? 
I  wanted  to  show  that  $\mathbb{R}\setminus \{0\}$  is  isomorphic  to $ \mathbb{R}\times \mathbb{Z}_2$.  

I first  tried  to  find  a  surjective  in  which  kernel  is  $\{0\}$.  But  I  failed,  so  I  was  a  little  confused  about  whether  these  two  sets  are  isomorphic  to  each  other  or  not.  So  can  it  be  first  shown  that  the  sets  cannot  be  non  isomorphic  or  something?
 A: Interpret $\mathbb{Z}_2$ as the multiplicative group $\{\pm 1\}$ and look at $$\mathbb{R} \times \mathbb{Z}_2 \longrightarrow \mathbb{R} \backslash \{0\}, \; \; (x, \pm 1) \mapsto \pm e^x$$
A: First, note that you want to prove that these two groups are isomorphic, but your post says sets. Remember, a group is a set together with an operation, and the operation is crucial to what the group is; you should make sure you understand what the operation is for each group before you proceed.
The operation for $\mathbb{R}\setminus\{0\}$ is presumably $\cdot$ (i.e., multiplication), so that for example, given $3,7\in\mathbb{R}\setminus\{0\}$, we have $3\cdot 7=21$, and the operation on $\mathbb{R}\times \mathbb{Z}_2$ is presumably addition in each coordinate, so that for example
$$(5.3,1)+(-1.1,1)=(4.2,0)$$
(since $1+1=0$ in the group $\mathbb{Z}_2$).
Now, understand your goal: you want to find an isomorphism between these two groups, i.e. a function $f:\mathbb{R}\setminus\{0\}\to\mathbb{R}\times\mathbb{Z}_2$ that is a group homomorphism and bijective.
To be a group homomorphism, for any $x,y\in \mathbb{R}\setminus\{0\}$ the function $f$ should satisfy
$$f(x\cdot y)=f(x)+f(y)$$
which is the well-known property that the logarithm function $\log:\mathbb{R}_{>0}\to \mathbb{R}$ satisfies. The function $\log$ is also a bijection from $\mathbb{R}_{>0}$ to $\mathbb{R}$. This gives you a hint that you can use the logarithm to define your desired group isomorphism $f$.
To deal with negative numbers, use the factor of $\mathbb{Z}_2$, and define $f:\mathbb{R}\setminus\{0\}\to\mathbb{R}\times\mathbb{Z}_2$ like this:
$$f(x)=\begin{cases}
\hphantom{-}(\log(x),0) & \text{ if }x>0\\
(\log(-x),1) & \text{ if }x<0
\end{cases}$$
This function is the inverse of the function from user362705's answer.
