What exactly is the unique union of a family of sets?

Question

It's quite an elementary question, but I couldn't find anything relevant to the query online. In Velleman's book "How To Prove It", 3.6.5, this excerpt can be found 1. It defines "a new set $∪!F$ by the formula $∪!F = \{x \mid ∃!A (A ∈ F ∧ x ∈ A)\}$". I frankly don't really know what it would mean. Is this even called a unique family of sets? Not sure, that's just what I believe it must be known as.

For example, if we let a family of sets $F = \{\{1, 2, 3, 4\}, \{2, 3, 4, 5\}, \{3, 4, 5, 6\}\}$, then the $∪F$ would obviously be $\{1,2,3,4,5,6\}$. What would $∪!F$ be? In my opinion, it would be the same, but I'm not sure since I don't fully understand the concept.

For some basic information about writing mathematics at this site see, e.g., here, here, here and here. — Another User, Dec 05 '23 at 15:33
This unique union is closer to being the set of unique elements of subsets of $F$, i.e. elements of exactly one of the subsets. — Henry, Dec 05 '23 at 23:38

score 9 · Accepted Answer · answered Dec 05 '23 at 15:31

The uniqueness is in the formula used to define the "unique union":

$\{x \mid ∃!A(A∈F \land x∈A) \}$.

This means that if an element $a$ belongs to e.g. two sets $A_1$ and $A_2$ of the family $F$ you have to discard it.

In your example above, only 1 and 6 will be in $\bigcup !F$.

What exactly is the unique union of a family of sets?

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