How do I write "each $a$ belongs to a certain $b$

Question

I have two types of elements: $a$ and $b$. Each $a_i \in A$ belongs to only one specific $b_j \in B$. e.g.

$a_1, a_2, a_3 \in b_1, \\ a_4, a_5 \in b_2$

I could use some help with writing this in a nice mathematical expression. So far I came up with

$a_i \in b_j \forall i,j$

But I think that this says "Each element $a$ is in every element $b$." How do I write it down such that each element $a$ belongs only to one element $b$ ?

(P.S. Set theory is really not my cup of tea, so I would gladly hear of any mistakes I'm making.)

Answer 1

Keep in mind that $\exists!$ stands for "there exists a unique," so you can write, for example,

\begin{equation*} \left(\forall a_{i}\in A\right)\left(\exists!b_{j}\in B\right)\left[a_{i}\in b_{j}\right]. \end{equation*}

You could also put more explicit emphasis on the indices. For example, if $i$ and $j$ are taken from the same index set $I$, you could write

\begin{equation*} \left(\forall i\in I\right)\left(\exists!j\in I\right)\left[a_{i}\in b_{j}\right] \end{equation*}