Suppose I have: $$A=\{(0,0)\}$$ $$B=\mathbb{R}^2$$
The dimension of Cartesian product $A \times B$ is $$ dim (A \times B)=0+2=2$$ because the dimension of one-point-set is $0$.
But if I take one element from $A \times B$, for example $(0,0,1,1)$, it is from $\mathbb{R}^4$, so the dimension of the product would be wrong.
Where am I making a mistake? Thank you for help.
The dimension is actually defined by number of linearly independent basis describing that space. The quadruples you defined can be described as following: $$X=\lbrace{(a,b,c,d)|a=b=0,c,d\in \Bbb R^2}\rbrace$$ A linearly independent vector basis as a subset of $\Bbb R^4$ can be defined as: $$v_1=(0,0,0,1)$$ $$v_2=(0,0,1,0)$$ Obviously $v_1$ and $v_2$ are linearly independent and the fully describe the defined subspace of $\Bbb R^4$. As an another example take a plane in $\Bbb R^3$. While the members of that plane are triplets that's trivial that a plane is 2-dimensional so our justification makes sense.