What do the symbols $\mathbb{Z}$ and $\mathbb{Z}_n$ mean on this discrete math problem? Currently I have come across a problem set which I cannot decipher or begin to ask or search because I do not know what kind of notation or problems these are.

Please circle the best description:
a. If $f:\mathbb{Z}\to\mathbb{Z}_7$ by $f(n)=[n]$, then $f$ is one to one. True False
b. $\mathbb{Z}_{12}$ has no zero divisors. True False
c. In $\mathbb{Z}_{15}$, $[3]$ is a zero divisor. True False
d. In $\mathbb{Z}_{12}$, $[3][4]=[1]$ (multiplication). True False

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My thoughts:
  a. $\mathbb{Z}$ implies $\mathbb{Z}$ ?
  b. $\mathbb{Z}$ represents $12$ but $3$ and $4$ are zero divisors. False
  c. $\mathbb{Z}$ represents $15$ in which divided by $3 = 0$. Thus True
  d. I have no idea
 A: $\mathbb{Z}$ refers to the set of integers (Wikipedia link),
$$\large \mathbb{Z}=\{\ldots,-2,-1,0,1,2,\ldots\}$$
$\mathbb{Z}_n$ for some number $n$, in this context, refers to the "integers modulo $n$" (Wikipedia link, notation is here but I recommend reading the full article), the set
$$\large\mathbb{Z}_n=\{[0],[1],\ldots[n-1]\}\\[0.1in]
{\small\text{also sometimes written as}}\;\;\large\{\overline{0},\overline{1},\ldots,\overline{n-1}\}$$
On each $\mathbb{Z}_n$, an addition and multiplication operation can be defined. For example,
$$\begin{align*}
\large[2] + [5] &\large = [3] \quad\text{in }\mathbb{Z}_{4} & \quad\large[2] \cdot [3] &\large = [2]\quad\text{in }\mathbb{Z}_{4}\\[0.1in]
\large[2] + [5] &\large = [0] \quad\text{in }\mathbb{Z}_{7} & \quad\large[2] \cdot [3] &\large = [6]\quad\text{in }\mathbb{Z}_{7}
\end{align*}$$
I would assume that, if these notations are showing up in your homework, they've been covered in class or are explained in the textbook - do you understand the examples above?

It seems  you're also confused by the notation for functions (Wikipedia link, notation is here but I recommend reading the full article). 
A: Your answers to b and c are correct, but c is not expressed well and I worry that shows a lack of understanding.  What do you mean by $\Bbb Z$ represents $k$?  For a, the arrow is not implies, it is showing the domain and range of the function $f$.  So you are taking $n \in \Bbb Z$ to $n \pmod 7$.  Is this one to one? For d, your answers to b and c seem to indicate you know $\Bbb Z_{12}$ is the integers $\pmod {12}$, so what is $3 \times 4$ in that ring?
