# 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:

1. $X \cap Y$
2. $X \cup Y$

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?

• Could you do something like $\pm$, but with union on top and intersection on bottom? Jun 21, 2017 at 2:27
• @probably_someone would that not imply that it is both (an intersection and union)? Jun 21, 2017 at 2:31
• Not always. Look at statements like "$a\lessgtr b$ implies $c\gtrless d$", which combined the two statements "$a<b$ implies $c>d$" and "$a>b$ implies $c<d$". Jun 21, 2017 at 2:40
• Thank you for your help @probably_someone. I'm not sure this is what I want though. I would like for the operator to be vague rather than concisely writing two similar statements. Jun 21, 2017 at 2:44

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 $$\bigcup_{i=0}^{n}\daleth_i(X_i,Y_i).$$

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 $X, Y, Z \in \mathcal{P}(S)$:

1. $X \circ X = X$
2. $X \circ Y = Y \circ X$
3. $X \circ (Y \circ Z) = (X \circ Y) \circ Z$
• This is almost exactly what I'd like to use. Do you have any suggestions on notation if I have a statement like $X \circ_{1} Y \circ_{2} Z$ where each $\circ_{i}$ is equal to $\cup$ or $\cap$? Especially if $i > 2$ Jun 26, 2017 at 13:53