# Theory of complexity – How to understand the notion of Boolean query via the definition of Immerman

Boolean queries are a special case of general queries. You can take $$tau$$ to be a practical vocabulary and choose a pair of $$tau$$-structures to represent "true" and "false".

For a concrete example, let's take a look at $$tau$$ to be the vocabulary with a single symbol of nullary relation $$T$$. Now, associate "true" with $$tau$$-structure $$mathfrak {T}$$ who has $$| mathfrak {T} | = emptyset$$ and $$T ^ mathfrak {T} = { langle , rangle }$$ (that is, the nullity relation that contains the empty tuple) and "false" with the structure $$mathfrak {F}$$ who has $$| mathfrak {F} | = emptyset$$ and $$T ^ mathfrak {F} = emptyset$$ (the nullary relation that does not contain the empty tuple).

Empty universes are a little awkward because they mean that $$( forall x , varphi) rightarrow exists x , varphi$$ is not a tautology. If you do not like this and / or you do not like zero relationships, make sure that $$T$$ unary and use a universe to an element.