Reading notations for definitions of sets

Question

I'm trying to understand some notation. I sort of have an idea of what this means and I would appreciate some feedback. $$T = \{(x,y,z) \mid \{x,y\} \in E,\ z = 5\}$$

I read this as: $T$ is a function which takes arguments $(x,y,z)$ where $x$ and $y$ are elements of $E$, and $z$ is always equal to $5$. $$S = \{(x,y,z) \mid g(x,y) = 4, 0<z<5\}$$

I read this as: $S$ is a function which takes arguments $(x,y,z)$ where $z$ is between $0$ and $5$, and then?

Is this on the right track? $$dV = T \cup B \cup S$$

So $dV$ is where $T$, $B$, and $S$ intersects?

Answer 1

$T$, $S$ and $dV$ are sets, not functions.

$\left\{ \right\}$ is standard notation for a set. The content of the set is defined within the $\left\{ \right\}$

The $|$ is to be read as 'such that'.

In one of your cases for example, S is the set of all elements $(x,y,z)$ such that the $x,y$, satisfy $g(x,y)=4$, where $g$ is a function of 2 variables, and $z>0$ and $z<5$

$\cup$ is the notation for set union.

$A\cup B$ is another set comprised of all the elements in $A$ and all the elements in $B$, and extends to multiple sets, i.e. you can write $A\cup B \cup C$ etc.

Answer 2

No, you're not. $T$ and $S$ are sets, not functions:

• $T$ is the set of all triplets of the form $(x,y,5)$;
• $S$ is the set of all triplets $(x,y,z)$ in which $g(x,y)=4$ and $0<z<5$.

Besides, $\cup$ stands for union, not intersection.