I would like to formally write something like this in the technical math report: "If set A is empty, after some operation, formula B will be just a boolean True formula"
Could someone formally write above statement, and point me the difference of "empty set of formula" \O and just "True formula" which is usually shown as \top ?
These two are fundamentally different:
$\emptyset$ is, well, an empty set. $\top$, however, is a statement.
$\emptyset$ is not a statement ... and $\top$ is not a set.
However, there is a close connection between $\emptyset$ and $\top$, which is probably what you have encountered somewhere and are trying to express. It's this:
The $\top$ can be seen as the generalized conjunction of an empty set of formula.
We can write that as:
$\top = \bigwedge \emptyset$
I also wonder if you are running into in the context of resolution, where we work with sets of formulas, called clauses.
A clause, however, corresponds to generalized disjunctions. Indeed, an empty clause $\emptyset$ corresponds to $\bigvee \emptyset$, which is equivalent $\bot$, not $\top$
To add to the confusion, though, a clause set is a set of clauses, and corresponds to a generalized conjunction. So, an empty clause set $\emptyset$ corresponds to $\top$
Finally, and more informally, any tautology like $\top$ effectively says 'nothing'. For example, the stereotypical tautology $P \lor \neg P$, which is equivalent to $\top$, is like saying: "It's going to rain later to day ... or not". Well, thanks for nothing! Tautologies can thus be said to have no information content. And, there is a loose connection with the empty set as well, because the empty set is, well, 'nothing'. As the above shows, though, you have to be very careful about this though. An empty set is still a set: a set that contains nothing, and $\top$ is still a statement: a statement that, in effect, says nothing.