Why are "~B" and "$\{x : x\notin B\}$" not the same thing? Trying to understand basic set theory. Why are they different: "~B" vs. $\{x:x \notin B\}$?  They both mean elements not in B. As long as that's satisfied, it should work, right?
How we use them in class:
$A\setminus B$ = {$x: x \in A \text{ and } x\notin B $}
$x \in A\setminus B$  iff $(x \in A) \land (x \notin B)$
Example:
Let B = {2, 4, 6, 8}.

the integers 1, 3, 5, 7, 9
the real numbers greater than 25
the function $f(x)=x^2+3$
the circle of radius 1 centered at the origin of the plane
the Empire State Building
my uncle Wilbur

Solution explains they are different:
$\{x:x \notin B\}$ contains all of the above.
1, 3, 5, 7 should be included in "~B", and perhaps real numbers greater than 25. But, most likely not the others. And certainly, knowing that my uncle Wilbur is not a member of the set B would contribute little to any discussion of B.
Question:
What's wrong with including the others? They are clearly not in B. I'm so confused right now.
 A: When taking the complement of some set we are (at least implicitly) working with some sort of universal set in the background. In your example, the universal set $X$ could be the set of all integers $\mathbb{Z}$, or the set of natural numbers $\mathbb{N}$, or the set $\{2,4,6,8\}\cup C$, where $C$ is the set of all cows. You can't meaningfully take the complement of a set unless you know what its background set is.
(In some sense, Russell's paradox is an extreme example of how things can break when you haven't specified your background set.)
A: As the comments indicate, the expression ~$B$ implicitly assumes a universe $U$ of possible elements some of which may be in the subset $B$.
Both the first two possible answers require a very improbable universe $U$. The third even more so. The last two are not even statements about mathematical objects, so make no sense at all.
A: Although it's true that the notion of "complement of a set" presupposes some notion of universal set, I would expect this universal set to include all the entities under discussion, even Wilbur, unless the contrary was explicitly stated in the question. And I don't consider "would contribute little to any discussion" relevant at all. Mathematical entities like sets don't exist solely to contribute to our discussions. As far as I'm concerned, that "solution", considered as mathematics, is hopelessly confused.
