@Zanna You could instead (or also) express the braced set notations with definite descriptions. Then those notations can be used to express "A ∪ B", "A ∩ B", "(a, b)", "A × B", "𝒫(S)", and so forth. (Which I've already done, above.) The main such notations are listing each element of a set, separated by commas, like {x₁, x₂, …, xₙ}, and a few forms of what is usually called set-builder notation, or abstraction, or occasionally comprehension.

I think (and hope, since I've been using it) that you're familiar with at least one form of set-builder notation already: {x | Fx} is the set of x for which Fx. I should say that I am using "Fx" quite informally here. In place of "Fx" could be any sentence in which "x" is free. It could have other free variables, and it need not even be an atomic sentence. For example: A ∩ B = {x | x ∈ A ∧ x ∈ B} And of course you do not need to name your variable "x".

@EliahKagan ∃y (Bvy ∧ Bwy)

@EliahKagan I realise that I don't know how to do any of these things at all

@Zanna Yes. For some y, Veromont has y as a state bird and Washington has y as a state bird.

@Zanna What's true of the empty set but nothing else?

Especially if the issue is that you're unaccustomed to writing definite descriptions, you may want, based on the sentence you wrote that expresses that Vermont and Washington share a state bird, to write a definite description for "the unique state bird of both Vermont and Washington."

Such a definite description will very strongly resemble that sentence, though of course it will not be the same sequence of symbols as it, since a definite description is a term, not a sentence. Also, that sentence does not, of course, guarantee that such a definite description succeeds; it would still fail if Vermont and Washington have multiple state birds in common.