How does one interpret the following set notation:
$$ \{(x_i,y_j): i=1,...,4,j=1,...,6\} $$
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1 Answer
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You can interpret it as a cartesian product, i.e. your set is just the cartesian product $X\times Y$ where $X=\{x_1,x_2,x_3,x_4\}$ and $Y=\{y_1,y_2,y_3,y_4,y_5,y_6\}$.
In general, in the notation you are citing, you can assume that any two variables are "independent", so long as there is no other information.
Note that this would also be true if the two variables had the same lenghts. That means that $\{(x_i,y_j): i=1,\dots, 4, j=1,\dots,4\}$ has $16$ elements, not $4$. If you specifically wanted to describe the set $\{(x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4)\}$, then you would need to explicitly explain that the variables $i,j$ are connected in some way, so you could write $$\{(x_i,y_j): i=1,\dots, 4,j=1,\dots,6 \land i=j\}$$ however this could also be written more compactly as $$\{(x_i,y_i)|i=1,\dots,4\}.$$
