What does the following mean translated to English from mathematical language?

Question

What does this mean in English language translated from mathematical text?

$$S=\left\{ \frac{1}{z} \mid z\in R\right\}$$

Answer 1

Since you have chosen "complex analysis" as a tag I guess that here we should see $z$ as a complex number (although this is not explicitly written).

Having said that, I guess that $S=\{\frac{1}{z}\;|\;z\in\mathbb{R}\}$ is the set of all the complex numbers $1/z$ such that $z$ is real number, that is the imaginary part of the complex number $z$ is zero.

In order to find a simpler description of the elements of the set $S$, note that, when $z\not=0$, if the imaginary part of $z$ is zero then also imaginary part of $1/z$ is zero. Why? Is the converse true?

Answer 2

This is standard set notation. In English it is read as "The set of numbers $\frac{1}{z}$ such that $z$ is a real number".

Answer 3

For all real numbers $z$ we compute the number $\frac{1}{z}$, whenever possible (so when $z \neq 0$) and put all these (and only these) in the set $S$. Note that the reciprocal of a real is also real, and every non-zero real can be written this way so $S$ is really the set of all nonzero real numbers, $\mathbb{R}\setminus\{0\}$, written in a weird way.