How can we use a cartesian product in the definition for bipartite graph when its elements are tuples and edges can be sets?

Question

I have learnt that in a bipartite graph, we can partition vertices into 2 sets $V_1$ and $V_2$, and $E \subseteq V_1 \times V_2$. However, the cartesian product is a set of tuples, and edges (in an undirected graph) are sets. So how can $E$ be a subset of this Cartesian product when a set is a different mathematical object than a tuple, so none of its elements are equal to any of the elements in the Cartesian product?

Answer 1

Core misconception is this :
" ... edges (in a directed graph) are sets ... "

Edges are not really sets ( though we might have some way to achieve that , that is not the concern here )
Edges are ... Edges !
All of those Edges are in some Set , though the Edges themselves are not Sets.

This is true :
" ... a set is a different mathematical object than a tuple ... "
though it is not going to impact the Case that Edges are not Sets.

Illustrative Example :

We have a Bi-Partite graph.

Nodes are in a Set $V=\{v_0,v_1,v_2,v_3,v_4,v_5,v_6,v_7\}$.
Here , neither $v_0$ nor $v_7$ is a Set. These are Nodes , nothing more , nothing less.

Edges are $E=\{ E_1 , E_2 , E_3 , \cdots \}$.
We can see the the Edges can be given in terms of the Nodes , like this :
$E=\{(v_0,v_4),(v_0,v_5),(v_1,v_4), \cdots \}$.
Here , too , neither $(v_0,v_4)$ nor $(v_3,v_5)$ is a Set. These are Edges , nothing more , nothing less.
We use "Names" ( like $E_1$ ) or "Constituents" ( like $(v_0,v_1)$ ) to refer to those Edges.

In "Constituents" notation , we can see that the tuples are all in $V_1 \times V_2$ where $V_1$ & $V_2$ are the Partitions & all "Directions" are from $V_1$ to $V_2$. In other words , $E \subseteq V_1 \times V_2$.

ADDENDUM :
Query is about Directed graphs.
In Case it is not Directed graph , either [[A]] we have to first fix the order of writing the tuples ( ordered Pairs , where $E_1=(v_1,v_4)$ ) in the Cartesian Product or [[B]] we have to change the Cartesian Product to something else to generate something else ( unordered Pairs , where $E_1=(v_1,v_4)=(v_4,v_1)$ ) to handle the 2 orders.