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I have a pair of variables $A_{i}B_{k}$, where $A_{i} \in \mathcal{A}$ and $B_{k} \in \mathcal{B}$. In the problem I am examining, I want to say that each such pair of variables, $A_{i}B_{k}$, is replaced with a corresponding set of elements, where each element is composed by the particular two variables $A_{i}B_{k}$ (with the exact indices, indicating the particular values of $A_{i}$ and $B_{k}$) and another variable $B_{l} \in \mathcal{B}$, such that each element is in the form $A_{i}B_{k}B_{l}$, for all possible values of $B_{l} \in \mathcal{B}$ (but for the particular values of $A_{i}$ and $B_{k}$).

For example, the pair $A_1B_a$ is transformed to a set $\{ A_1B_aB_a, A_1B_aB_b, A_1B_aB_c,..., A_1B_aB_z \}$, i.e. the indices of the first two variables always remain the same.

I would like to express this "transformation" of each pair of variables $A_{i}B_{k}$ using the set builder notation, but I am not sure how to do this properly. Should I say that:

"Each pair of variables $A_{i}B_{k}$, where $A_{i} \in \mathcal{A}$ and $B_{k} \in \mathcal{B}$, is replaced by a corresponding set $\{ A_{i}B_{k}B_{l}| B_{l} \in \mathcal{B} \}$."

Is there a better way to indicate that in each term $A_{i}B_{k}B_{l}$ in the set, the first two variables (i.e. their indices) remain the same, whereas the third variable $B_{l}$ takes all the possible values in $\mathcal{B}$?

Asaf Karagila
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E-O
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1 Answers1

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You could indicate it as a function, i.e. write: $$ f(A_{i}B_{k}) = \{ A_{i}B_{k}B_{l}| B_{l} \in \mathcal{B} \} $$

And then state that each $A_iB_k$ is replaced by $f(A_iB_k)$ where $f$ is as defined above.

This makes the constancy of $A_iB_k$ clear in the set; and also that the transformation takes place for each $A_i B_k$.

whoisit
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