With the relations $A\subseteq\mathbb{N}\times\mathbb{N}$ with $A=\{(1,1),(2,2),(1,2),(2,1)\}$ and $B\subseteq \mathbb{N} \times \mathbb{N}\times \mathbb{N}$ with $B=\{(1,2,3),(4,5,6),(1,1,1),(1,2,3)\}$.
Define the elements of the set
$M=\{(a_{0},a_{1},b_{0},b_{1},b_{2})|(a_{0},a_{1}) \in A \wedge |(b_{0},b_{1},b_{2})\in B \wedge a_{0}=b_{0}\}$
To define the cross product $M=A \times B$, I use only the elements of the sets A and B, under the condition that $a_{0}$ and $b_{0}$ are the same.
$M=\Bigl\{(1,1),(1,2)\Bigr\}\times\Bigl\{(1,2,3),(1,1,1),(1,2,3)\Bigr\}$
My two possible solutions would be:
$M_1=\left\{\Bigl((1,1,1,2,3),(1,1,1,1,1),(1,1,1,2,3),(1,2,1,2,3),(1,2,1,1,1),(1,2,1,2,3)\Bigl)\right\}$
$M_2=\left\{\Bigl(((1,1),(1,2,3)),((1,1),(1,1,1)),((1,1),(1,2,3)),((1,2),(1,2,3)),((1,2),(1,1,1)),((1,2),(1,2,3))\Bigl)\right\}$
My question is which of these both solutions is true $M_1$ or $M_2$? Or are both false, if that's the case how shall approach to get the right solution?
The correct form is
$$M_2=\Bigl\{((1,1),(1,2,3)),((1,1),(1,1,1)),((1,1),(1,2,3)),((1,2),(1,2,3)),((1,2),(1,1,1)),((1,2),(1,2,3))\Bigr\}$$