From what I know, the union and intersection class of any class $A$ is defined as the following;
$$\bigcup A = \{x:x \in y \text{ for some } y \in A\}$$
$$\bigcap A = \{x:x \in y \text{ for every } y \in A\}$$
I am new to set notation so I am having trouble figuring out what it is. If I have a class $A = \{1,2,3\}$ what will its union and intersection class be?
I worked it out and from my understanding, the former is any subset of $A$ and the latter is simply the set of all subsets of $A$, but I don't feel that's right; it seems unnecessary since they are just repeating the definitions of a subset and a powerset.
Best Answer
When we speak of $\bigcup A$ and $\bigcap A$, we require all the elements of $A$ to themselves be sets.
Depending on the precise foundational theory you're working in, one of a few things can happen.
A 1-sorted theory is, as the name suggests, a logical theory in which there's only 1 sort of thing. In ZF, the only sort of thing that exists is sets. All the "things" discussed in ZF are sets.
In this case, all elements of $A$ are automatically sets, so the notation always makes sense.
In ZF, one typically defines $0 = \{\}$, $1 = \{0\}$, $2 = \{0, 1\}$, $3 = \{0, 1, 2\}$. In this case, we see that $\bigcup \{1, 2, 3\} = \{0, 1, 2\} = 3$. In fact, if $A$ is a finite set of natural numbers, $\bigcup A$ will be the largest element of $A$ and $\bigcap A$ will be the smallest element.
However, there is a small wrinkle with ZF-like theories. The wrinkle is that $\bigcap A$ is not defined when $A = \emptyset$. This is because according to the definition of $\bigcap \emptyset$, we would expect absolutely anything to be an element of $\bigcap \emptyset$ - that is, $\bigcap \emptyset$ would be the set of all sets. But we know there is no set of all sets, since if there were such a thing we would have Russell's Paradox.
In NBG set theory, the things one quantifies over are "classes". There is, for example, a class of all sets. A "set" is a special kind of class which is "small enough" in some sense.
This allows a different approach to axioms than ZF. In ZF, there is an axiom saying that for all $a$, there is a set $P(a)$ of all subsets of $a$. In NBG, given a set $a$, you can immediately define the class $P(a) = \{b \mid b \subseteq a\}$; the axiom says that $P(a)$ is a set.
In NBG set theory, the intersection $\bigcap A$ is always defined. But if $A$ is empty, the intersection will not be a set. It will instead be a "proper class" - that is, a class which is not a set.
In this version of events, the "things" one quantifies over might be "atoms". An atom is something which is not a set/class and which is thus not defined solely in terms of its elements (atoms are generally taken to have no elements at all). In such a set theory, we would only discuss $\bigcap A$ and $\bigcup A$ when all elements of $A$ are sets (not atoms), though (assuming we adopt the convention that atoms have no elements) the notation $\bigcap A$ and $\bigcup A$ makes sense even when $A$ has atoms - in this case, $\bigcup A = \bigcup \{x \in A \mid x$ a set$\}$ and $\bigcup A = \emptyset$.
In type theory, everything has a "type". It does not make sense in type theory to take the union or intersection of types.
However, given a type $T$, one can (sometimes) form the type $P(T)$, which is (roughly) the type of all subsets of $A$. Given some $A : P(P(T))$, one can define $\bigcup A = \{x : T \mid \exists y \in A (x \in y)\}$ and $\bigcap A = \{x : T \mid \forall y \in A (x \in y)\} : P(T)$.
Here, $t : T$ means "$t$ has type $T$".
In this account, $\bigcap A$ is actually well-defined even when $A = \emptyset$.
In type theory, trying to take the union $\bigcup \{1, 2, 3\}$ would be a "type error". One cannot even discuss it. This is because $1, 2, 3$ are not sets.
The type theoretic/category theoretic account of intersections and unions is a bit more subtle when one does not have "power set" types. I will not go into that here.