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

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. Dec 5, 2023 at 15:33
• It might help to to try drawing this out in a Venn diagram. Dec 5, 2023 at 18:04
• You can think of it as a generalized xor. Dec 5, 2023 at 19:21
• 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. Dec 5, 2023 at 23:38

$$\{x \mid ∃!A(A∈F \land x∈A) \}$$.
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 $$\bigcup !F$$.