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How can I state using Set Notation that each element $n$ in the set $A$ contains a unique set $B$ ?

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$n$ contains a set $B$? Or $n$ belongs to $B$? – Sigur Feb 3 '13 at 20:30
@Sigur you probably mean $B \in n$ instead of $n \in B$. – Trevor Wilson Feb 3 '13 at 21:13
@TrevorWilson, my comment was based on a deleted comment by the OP. – Sigur Feb 3 '13 at 21:26
up vote 2 down vote accepted

$$\forall n\in A\,\,\exists ! B\,\,,\,B\,\,\text{is a set and}\,\,\,B\subset n$$

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What does $$ B\subset n$$ mean in this context? That $B$ has fewer elements than the set $n$? – Inge Eivind Henriksen Feb 3 '13 at 20:53
it means that B is a subset of n. – pascalhein Feb 3 '13 at 21:14
If $A$ is a set containing cars and $B$ is a set containing parts , then how can $B$ be a subset of $n$ ? – Inge Eivind Henriksen Feb 3 '13 at 21:22
The above is exactly what you say:"Each element in A contains a unique set B" . As other comments say, I highly doubt this is what you *actually * meant, but that's not my problem. – DonAntonio Feb 4 '13 at 2:33
+1 for nice convincing explanation in the last comment. – Babak S. Feb 4 '13 at 9:17

If you mean that $B$ is an element of each $n\in A$, you can write $B\in\bigcap A$: for any collection $A$ of sets, by definition $x\in\bigcap A$ if and only if $x\in n$ for each $n\in A$. If you mean that $B$ is a subset of each $n\in A$, then each element of $B$ is an element of $\bigcap A$, and you can write $B\subseteq\bigcap A$.

There are of course more complicated ways to express the same things.

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