Boolean and equivalent to summation

Question

Is there a mathematic symbol to express the application of AND operator to a set of booleans, that returns true only of all booleans in the set are true. Something like the summation operator on a set of integers.

Answer 1

As is common for most binary operators outside of multiplication and addition, which use $\prod$ and $\sum$ to stand for the iterated version of $\cdot$ and $+$, most iterated versions of binary operations simply use an enlarged version of that symbol to represent it. For example, the union of a collection of sets is given by $\bigcup_{A\in\mathcal{A}}{A}$.

Given a binary operator $\oplus:A\times A\rightarrow A$ that is associative, the iterated version of $\oplus$ is defined inductively by $\bigoplus_{i=1}^{n+1}{a_i}=\bigoplus_{i=1}^n{a_i}\oplus a_i$ and satisfies the usual properties of

1. $\bigoplus_{i=1}^1{a_i}=a_1$,
2. $\bigoplus_{i=1}^2{a_i}=a_1\oplus a_2$, and
3. $\bigoplus_{i=1}^n{a_i}\oplus \bigoplus_{i=n+1}^m{a_i}=\bigoplus_{i=1}^m{a_i}$.

It is important to notice that this can only be used to define $\bigoplus$ for finite collections of elements; for operations like the union, intersection, or Cartesian product, other methods are used to define them (in the first two cases, specifically, the Axiom Schema of Restricted Comprehension and quantifiers).

For your specific binary operator in mind, $\wedge$, or conjunction, we do the same to get $\bigwedge_{B\in\mathcal{B}}{B}$. Similarly, for disjunction, $\vee$, we do the same to get $\bigvee_{B\in\mathcal{B}}{B}$.

(Note that the well-definedness of such an operation, at least in finitary logic, where you can only have sentences of finite length.)