How to visualize the Cartesian product of a set and an interval For example, suppose I have the following: $[1,5] \times \{-2,2\}$. 
I know that $[1,5]$ is an interval and the set $\{-2,2\}$ is a set containing two elements, but I am getting confused with the elements in the set--do the elements in the set mean lines? Like $y=2$ and $y=-2$?
$x=[1,5]$, $y=\{-2,2\}$?
 A: Consideration: It may help to think about your more general problem of drawing Cartesian products in terms of intervals first because a simple $2$-element set will be more restrictive  (and thus easier to visualize). 
Example 1: Consider the sets $A=[1,4)$ and $B=(2,4]$. Then $A\times B$ depicts a rectangle in $\mathbb{R}^2$ defined by 
$$
A\times B = \{(x,y) : 1\leq x< 4\;\text{and}\, 2<y\leq 4\}.
$$
Example 2: Consider the sets $A=(1,5], B=[1,4), C=[4,7), D=(2,5)$. Then we can actually visualize the identity
$$
(A\times B)\cap (C\times D) = (A\cap C)\times (B\cap D)
$$
as a rectangle in $\mathbb{R}^2$ defined by
$$
\{(x,y) : 4\leq x\leq 5\;\text{and}\; 2<y<4\}.
$$
Your problem: If you are able to see how those examples worked, then it should not be too much of an issue to see how, when $A=[1,5]$ and $B=\{-2,2\}$, we have
$$
A\times B=\{(x,y) : 1\leq x\leq 5\;\text{and}\; y\in\{-2,2\}\}.
$$
Thus, we will have two lines in $\mathbb{R}^2$ given by $y=-2$ and $y=2$, where both of these lines are restricted to the interval $1\leq x\leq 5$. Hence, for your sets $A$ and $B$, your set $A\times B$ will look like this:

A: Each element of $[1,5]\times \{2,-2\}$ is a pair $(x,y)$ where $x\in [1,5]$ and $y\in \{-2,2\}$.
One simple way to visualize this is to view each element as a point in the plane.  In this case you just have two separate copies of the interval, one at $y=-2$ and the other at $y=2$.
A: It looks like you already understand. The Cartesian product is defined as 
$$A\times B:=\{(a,b)\mid a\in A,\; b\in B\}$$
So in this case it would consist of all the points (coordinates in $\mathbb{R}^2$) $(x,-2)$ and $(x,2)$ where $1\leq x\leq 5$. Basically these are two line segments, as I think you've already said correctly.
