What does the notation $z\in\mathbb{C}\backslash\mathbb{R}$ mean? I know that $z\in\mathbb{C}/ \mathbb{R}$ means that the domain is the complex plane with the real line removed.
What does the notation $z\in\mathbb{C}\backslash \mathbb{R}$ mean?
EDIT:
Turns out that I got things backwards. If $\mathbb{C}\backslash \mathbb{R}$ is the set difference, what does $\mathbb{C}/ \mathbb{R}$ actually mean?
 A: $z \in \mathbb{C} \setminus \mathbb{R}$ means that $z$ is a complex number that is not a real number.  I.e., any number of the form $a+bi$ where $b \not= 0$.
The "backslash" $\setminus$ is the set-difference or set-minus operation.  In general $A \setminus B$ is the set of all $x \in A$ such that $x \not\in B$.
The "forward slash" $/$ is a quotient operator.  $\mathbb{C}/\mathbb{R}$ would be the set of all cosets of $\mathbb{R}$ in $\mathbb{C}$; these cosets each contain complex numbers whose imaginary components are equal.
A: Actually, $\setminus$ is the set difference.
A: $\mathbb{C}/\mathbb{R}$ is the set of subsets of $\mathbb{C}$ of the form $$z + \mathbb{R} = \{z + r : r \in \mathbb{R} \}$$ where $z$ is a given complex number. Two sets $z+ \mathbb{R}$ and $w+ \mathbb{R}$ might be equal, even if $z \neq w$.  You can show that two of these subsets $z + \mathbb{R}$ and $w+ \mathbb{R}$ are equal if and only if the difference $z - w$ is a real number, i.e. if $z$ and $w$ lie on the same horizontal line parallel to the real axis.
A: It depends on the context (and the course) in which you are working.
As many people have mentioned already, if you are treating $\mathbb{C}$ and $\mathbb{R}$ as vector spaces (in an abstract algebra or linear algebra class, for example), then $\mathbb{C}/\mathbb{R}$ is the set of cosets of $\mathbb{R}$ in $\mathbb{C}$. 
If, however you are in the context of topological spaces (as in a topology course), then $\mathbb{C}/\mathbb{R}$ is the space of all complex numbers viewed as points, where the line of real numbers has been collapsed down to a single point.You could draw the space as an hourglass where the middle point of the glass is the collapsed real line. The important difference here is that in topology, saying that all the real numbers are going to be treated as the same point does not cause any of the other complex numbers to be equal to each other. In the algebraic quotient, it does.
in the algebraic quotient we would have $4 + 7i = 9 + 7i$ since they differ by the real number 5. In the topological quotient, these two numbers would stay unequal because they are not real numbers.
