Set in a set notation

I'm not sure how to clearly express this informally, but

$A$ is a set containing cars, and $B$ is a set containing parts. But it is also so that $A_1$'s $B$ set is not the same set as $A_2$'s $B$ set, etc.

I could also use the following example to explain what I mean; $A$ is set of moms and $B$ are sets of children that a specific mom in the $A$ set has. Lets say $A_1$ is 'Kari' and $A_2$ is 'Lisa', 'Kari' is the mother of 'Karl' & 'Tia', while 'Lisa' is the mother of 'Bill'. The $B$ set does not contain all the children regardless of mother, but each mother has her own $B$ set.

How can I state using something like this using Set Notation ?

-

It seems that what you are talking about is a set-function. It is a function which takes an element from $A$, say a mother, and returns the set of children of the said mother.

One way to write it would be $\{B_a\mid a\in A\text{ and } B_a\text{ is the set of children of }a \}$.

-
That seems very clear, thanks! –  Inge Henriksen Feb 4 '13 at 11:45
@Asaf, don't you just mean to define $B_a$ to be the set of children of $a$? –  Trevor Wilson Feb 4 '13 at 18:15
@Trevor: Mean where? –  Asaf Karagila Feb 4 '13 at 18:18
@Asaf I'm just not sure what the set you wrote is supposed to be. Is $a$ bound or free? –  Trevor Wilson Feb 4 '13 at 18:19
@Asaf That is a good point. It just looked strange to me in this context. If I dealt with sequences indexed by mothers more often then I probably would not bat an eye at it :) –  Trevor Wilson Feb 4 '13 at 18:33

For every mother $a \in A$, you could define the notation $B_a$ to be the set of children of $a$—in symbols, $B_a = \{b : b \text{ is a child of } a\}$.

Or if you have enumerated the set $A$ of mothers as $\{a_1, a_2, \ldots\}$ then instead of writing $B_{a_i}$ for the children of the $i^\text{th}$ mother $a_i$, you could just write $B_i$ instead. So $B_1$ denotes the set of children of $a_1$, and $B_2$ denotes the set of children of $a_2$, and more generally $B_i = \{b : b\text{ is a child of }a_i\}$.

-