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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.)

Asaf Karagila
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Chris_abc
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1 Answers1

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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*}

kandb
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  • This looks perfect. The "there exists a unique" operator seems very useful in this case :) – Chris_abc Feb 09 '23 at 09:58
  • One last question, why do you use the brackets in $[a_i \in b_j]$? Could I also write $\forall a_i \in A, \exists ! b_j \in B s.t. a_i \in b_j$ ? the "s.t." notation seems to be used often in my university. – Chris_abc Feb 09 '23 at 10:02
  • @Chris_abc absolutely! I only used the brackets and parentheses to visually separate the different quantifiers and the statement they modify--it's just a matter of choice, as long as it's clear what you mean =) – kandb Feb 09 '23 at 10:03