Let's for the sake of simplicity only focus on tuple relational calculus.
Every relational algebra query can be broken down into the 5 atomic operations - projection, selection, set union, set difference, cartesian product (I'm excluding renaming as it is not possible to do in relational calculus).
Let $P(x)$ be an arbitrary predicate and $r$, $s$ relations with attributes $a_1,\dots,a_n$ and $b_1,\dots,b_n$ respectively.
For the 5 atomic operations, we have a conversion: $\newcommand{\Set}[2]{% \{\, #1 \mid #2 \, \}% }$ \begin{align} \Pi_{a_{1},\dots,a_{n}}(r) &\iff \Set{t}{\exists p \in r \left(\bigwedge\limits_{i=1}^n t.a_i = p.a_i \right)}, \\ \sigma_{P(x)}(r) &\iff \Set{t}{t \in r \land P(x)},\\ r \cup s &\iff \Set{t}{t \in r \lor t \in s}, \\ r \setminus s &\iff \Set{t}{t \in r \land t \notin s}, \\ r \times s &\iff \Set{t}{\exists p \in r \; \exists q \in s \left( \bigwedge\limits_{i=1}^n t.a_i = p.a_i \land \bigwedge\limits_{i=1}^n t.b_i = p.b_i \right)}. \end{align}
But we only used $r$ and $s$ which are relations. What if we have an expression in a form of $Q_{1} \bigoplus Q_{2}$ where $Q_1$, $Q_2$ are also queries?
Is it possible to write a general formula for the 5 main operations while only using the most general $Q_1$, $Q_2$?
