I have a dynamic situation consisting of two sets. This situation occurs in my work so it’s not a theoretical or ‘strawman’ argument.
Let $a\in A$ and $b\in B$. These sets could be joint or disjoint depending on whether they share a common element (or elements) which is unknown until the elements can be matched (i.e. there is no prior assumption as to whether or not there are any common elements).
I am trying to write the intersection of these sets using set builder notation. At this stage, with my 'working' knowledge of set theory, I have come up with the following:
$$A \cap B = \{a,b \;|\;a \in A \; and \; b \in B,\; ∃a=b\}$$
However, this does not satisfy my situation as the existential quantifier (∃) means “there exists”, “there is at least one”, or “for some”. In my dynamic situation the appropriate quantifier needs to mean “there $\pmb{may}$ exist”, “there may be at least $\pmb{none}$”, or “for $\pmb{maybe}$ some”. Formalising the notation is important in order to help identify when the sets are not mutually exclusive as a stepping stone to further analysis.
Assuming it can be done, what is the best – and simplest – way to write this dynamic situation using set builder notation?
It is simply :
The formula reads : "the intersection of two sets $A$ and $B$ is the set of all and only those elements that belong to both $A$ and $B$."
The role of the set-builder notation $\{ x \mid \text { condition on } x \}$ is to specify that the set is formed by all and only those elements that satisy the condition.
To say that $A$ and $B$ are disjoint, we may use : $A \cap B = \emptyset$ that reads : "there is no common element between $A$ and $B$.
In formula : $\lnot \exists x \ (x \in A \cap B)$.