I am taking a discrete mathematics course in the spring and in an attempt to fully understand the material I am reading ahead. I came across this statement Let $\mathbb{R}$ denote the set of all real numbers. Describe $\mathbb{R} × \mathbb{R}$, and it got me thinking. I did not quite understand it at first until I saw the answer and re-read it a few times. For me the ah-ha moment was after reading it a second time and seeing the answer. So I included the answer to share with the community because maybe someone else out there could benefit in the same way I did when I saw the answer.
After completing my research on this site on how to describe a Cartesian Product of the form $\mathbb{R} × \mathbb{R}$, I was only able to come up with descriptions that a.) did not apply directly to what I was looking for and b.) were over-complicated in there response.
So again if $\mathbb{R}$ is said to denote the set of all real numbers. Describe $\mathbb{R} × \mathbb{R}$.
Formal Defintion:
Given sets $A$ and $B$, the Cartesian product of $A$ and $B$, denoted $A × B$ and read “$A\: \mbox{cross} \: B$,” is the set of all ordered pairs $(a, b)$, where $a$ is in $A$ and $b$ is in $B$. Symbolically: $A × B = \left\{(a, b) \: | \:a \: \in \: A \: \mbox{and} \: b \in B\right\}$
Thus we have:
$\mathbb{R} × \mathbb{R}$ is the set of all ordered pairs $(x, y)$ where both $x$ and $y$ are real numbers. If horizontal and vertical axes are drawn on a plane and a unit length is marked off, then each ordered pair in $\mathbb{R} × \mathbb{R}$ corresponds to a unique point in the plane, with the first and second elements of the pair indicating, respectively, the horizontal and vertical positions of the point.
The term Cartesian plane is often used to refer to a plane with this coordinate system.
Image and source Credit: Discrete Mathematics and Its Applications, 4th Edition, Epp