Notation for power set of coupled pairs

Question

Consider sets $\mathcal{B} = \{1,\ldots,B\}$ and $\mathcal{A}_b = \{1,\ldots,A_b\}$ for each $b\in\mathcal{B}$.

Now consider pairs of the form $(b,a)\in\mathcal{B}\times\mathcal{A}_b$. I want to write the set of all such pairs (i.e. the powerset). I am stuck as to how to write this. Any ideas?

Answer 1

I'm not aware of any better method than $$\{(b,a): b \in B, a \in \mathcal{A}_b\}$$ although if your audience is type-theoretically inclined, they may recognise the dependent sum type $$\sum_{b : B} A_b$$

Answer 2

$\cup_b$ B×$A_b,$
the collection of all such pairs, is not a power set.