Is there a standard way to write a set operation which may be either the union or intersection operator? I have set $X$ and $Y$. I want to write a statement generalizing the statements:


*

*$X \cap Y$

*$X \cup Y$


Something like $X~C_{i}~Y$ where I might say that $C_{1} = \cap$ and $C_{2} = \cup$.
Is there a standard way of writing a statement like this?
 A: I started typing too much for a comment, so I'll just leave this here as an answer:
I cannot say there is a standard, as I have never seen this before. I would have no issues reading what you've written. It looks good to me! 
Assuming you want to be able to "flip a switch," and just to offer a suggestion, you might try a function notation. Then we could manipulate things like:$$\daleth_i(X,Y)=\begin{cases}X\cup Y, &i \text{ is even}\\X\cap Y, &i \text{ is odd}\end{cases}$$
I chose daleth simply because it isn't used for much else, and $C(X)$ is sometimes used to mean complement. Not totally sure what your goal is, but this could allow for things like
$$\bigcup_{i=0}^{n}\daleth_i(X_i,Y_i).$$
A: I would write something along these lines:
Let $S$ be a set and let $\circ$ be a binary operator on $\mathcal{P}(S)$.
Proposition. If $\circ$ is equal to $\cup$ or to $\cap$, then the following properties hold for all $X, Y, Z \in \mathcal{P}(S)$:


*

*$X \circ X = X$

*$X \circ Y = Y \circ X$

*$X \circ (Y \circ Z) = (X \circ Y) \circ Z$

