Can't understand meaning of symbols of line segment formula: $\{ (x, y): x = a \;\&\; b \leq y \leq b + d^2 \}$

Question

My task is following:
find the number of intersection points of cirle $\mathbf{C} = \{ (x,y): x^2 + y^2 = r^2 \}$ and line segment (part that I don't understand):
$$ \mathbf{D} = \{ (x, y): x = a \;\&\; b \leq y \leq b + d^2 \} $$
Why is there $\&$ between $a$ and $b$? What do $a$, $b$ and $d$ stand for? ? Are they used in some kind of line segment formula? If they do, which one?

Answer 1

The ampersand $\&$ means "and". So the set $\mathbf{D}$ consists of ordered pairs $(x, y)$ such that $x = a$ and $y$ is between $b$ and $b + d^2$. This is a segment of the vertical line $x = a$ whose lowest point is at $y=b$ and whose highest point is at $b + d^2$.

By the way, it is much more common in defining a set this way to use either the word "and" or the logical conjunction operator $\land$ (which means "and"), perhaps with parentheses to clarify: $$ \mathbf{D} = \{ (x, y): (x = a) \;\text{and}\; (b \leq y \leq b + d^2) \} $$ or $$ \mathbf{D} = \{ (x, y): (x = a) \;\land\; (b \leq y \leq b + d^2) \}. $$

You can play around with the parameters $r$, $a$, $b$, and $d$ as sliders here to get a feel for the types of solutions (intersection points you can get): $0$, $1$, or $2$.