Does $\mathbb{N} \times \mathbb{R}$ make sense?

Question

I know that it is possible to have a Cartesian product for two sets of different cardinalities if the cardinality of both sets is finite. Taking $A = \{1,2\}$ and $ B = \{3,4,5\}$, $A \times B = \{(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)\}$. But what if we have one set where it has uncountably infinite elements and one with countably infinite elements? Does $\mathbb{N} \times \mathbb{R}$ make sense? I can understand it geometrically as the graph shown below but I'm not well versed enough with set theory to know if rigorously speaking this creates any problems/is allowed.

Answer 1

You can take the Cartesian product of any two (or even more) sets whatsoever; there are no cardinality restrictions of any kind.

Answer 2

Yes, for most set theories, $A \times B$ is always a valid expression. In your case, you've got it exactly correct with the graph. The graph of $\mathbb{N}\times\mathbb{R}$ is in fact the collection of points $(n, r)$ where $n$ is a natural number and $r$ is a real number.

If you want to be super rigorous about it, in ZF the existence of $A \times B$ follows from the axioms of pairing, union, power set, and specification on the formula "$\exists a\exists b[a \in A \wedge b \in B \wedge x=\langle a, b\rangle]$."