How to tackle an intersection of two sets with different variables? { x ∈ R : $-2 ≤ x ≤ 1$ }∩{ y ∈ R : $0 < y < 2$ }

Question

I just started picking up the intros in set theory and kinda confused by this as I am trying to determine the elements of the operation: { x ∈ R : $-2 ≤ x ≤ 1$ }∩{ y ∈ R : $0 < y < 2$ }

It just confuses me to think of them as a normal two sets getting intersected as I don't know how to apply the intersection definition when the variables aren't the same which leads me to think of it as an operation of cartesian points set or is the variable change is just an arbitrary thing and I should just focus on the elements?

Thank you!

Answer 1

There is no issue here.

The set-builder notation used in the formula $\{ x \in \mathbb R \mid \text {blah blah} \}$ means :

the set of all and only those (real) numbers such that : $\text {blah blah}$.

Thus, nothing changes in "calling" them $x$ or $y$.

So, you are simply asked to "compute" the intersection of two sets: the set all the reals belonging to $[−2, 1]$ and the set of all the reals belonging to $(0, 2)$.

Answer 2

The symbol $x$ in $\{x: -2\leq x \leq 1\}$ is a 'dummy' variable. The set remains the same if you change $x$ to any other symbol. The desired intersection is $[-2,1] \cap ((0,2)=(0,1]$.