# Notation for non-empty subset

To denote non-empty subsets, I repeatedly find myself writing $A\subset S, A\neq \emptyset$. Is there any established shorthand for this, you know, like $A\subset S$ can be seen as a shorthand for $A\subseteq S, A\neq S$?

• But it’s not the same?! Commented Oct 21, 2014 at 17:40
• I agree, it is certainly related (see goblin's answer), but not exactly the same thing. Commented Oct 21, 2014 at 19:28

Try $\emptyset \neq A \subset S$.

• I don't like this kind of thing (not downvoting or anything, just expressing my opinion). There's nothing more confusing than reading: "Now consider $\emptyset \neq A \subset S$." Commented Oct 21, 2014 at 17:32
• Actually I agree. Personally whenever possible I prefer sentences: "Let $A$ be a nonempty subset of $S$." Commented Oct 21, 2014 at 22:20

If I recall correctly, $A \in \mathcal{P}_{\geq 1}(S)$ is somewhat standard notation for "$A$ is a non-empty subset of $S$."

More generally, I think that $\mathcal{P}_{\geq \kappa}(S)$ is sometimes used to denote the collection of all subsets of $S$ with cardinality greater than or equal to $\kappa.$

• But it’s harder to read like $\subset$ … Commented Oct 21, 2014 at 17:42
• @Ronny, I know what you mean. Another option would be: "Consider non-empty $A \subseteq S$." Commented Oct 21, 2014 at 17:45
• Well, that wouldn't work in, say, indexing sets using a summation sign like $\sum_{A\subset S} f(A)$. Commented Oct 21, 2014 at 19:30

Define $A \subset_\emptyset B \Leftrightarrow (A \subset B \wedge A \neq \emptyset)$. Then use $\subset_\emptyset$ instead of $\subset$.

• Thanks, I am aware of such possibilities. I was just wondering if there was some established standard of doing that, you know, $\subset$ with a dot in it or something. Commented Oct 21, 2014 at 17:25
• Ah, okay, not as far as I know – sorry. Commented Oct 21, 2014 at 17:27

Probably not, notations differ from author to author. Not even $A\subset S$ is unambiguous - it can mean proper subset or just subset. This means that if you're going to use a shorthand notation you would probably need to define the meaning of it. The mentioned $\mathcal P_{\ge 1}$ for example is nothing I've encountered, and the same applies to $\subseteq_{\emptyset}$.

Like it or not the notation $\emptyset\ne A\subset S$ or even $A\subset S, A\ne0$ is the closest thing to something that would be immediately understood and it's almost as compact as $A\subset_{\emptyset}S$, or $A\in\mathcal P_{\ge1}(S)$.

Normally you would have it in text and then you could write it out "non-empty $A\subset S$". As pointed out you should consider the context in the case as well. If you want to use $\emptyset\ne A\subset S$ you should use it in a context that calls for a statement, for example "assume that $\emptyset\ne A\subset S$".