# Notation for partitions of sets

A partition of a set $$X$$ is a set of non-empty subsets of $$X$$ such that every element $$x$$ in $$X$$ is in exactly one of these subsets (i.e., $$X$$ is a disjoint union of the subsets).

Equivalently, a family of sets $$P$$ is a partition of $$X$$ if and only if all of the following conditions hold:

• The family $$P$$ does not contain the empty set (that is $${\emptyset \notin P}$$).

• The union of the sets in $$P$$ is equal to $$X$$ (that is $${\bigcup_{A\in P}A=X})$$. The sets in $$P$$ are said to cover $$X$$.

• The intersection of any two distinct sets in $$P$$ is empty (that is $$(\forall A,B\in P)\;A\neq B\implies A\cap B=\emptyset$$). The elements of $$P$$ are said to be pairwise disjoint.

1. Is there a concise way of expressing mathematically that $$P$$ is a partition of $$X$$?

2. If $$x$$ is a member of $$X$$ then $$x$$ must necessarily be a member of one and only one of the sets in $$P$$. How can I express "the set in $$P$$ that $$x$$ is member of"?

3. How can I express a function that associates members of $$X$$ with members of $$P$$? (In other words, a mapping from members of $$X$$ to the partition it belongs to.)

Please try to answer question 2 without being biased by question 3.

EDIT: As pointed out in answer and comment, I am already expressing "$$P$$ is a member of $$X$$" mathematically, in unambiguous English. My question was written under the false presumption that mathematical notation means symbolic notation. I let the original wording stand, as not to render the already given answers/comments meaningless. But I was looking for a more symbolic notation of the given predicates.

• You've already expressed all those things concisely and mathematically in your own question. Writing things unambiguously in English is itself a way to express mathematics. Feb 3 '20 at 10:59
• As an additional comment based on your edit: in fact, unambiguous and clear English is often preferred to unnecessary symbolic notation. Using symbolic notation to express elementary ideas more often than not obscures the thing you want to talk about. For example, I can express that $P$ is a partition of $X$ by stating that $P\subseteq\mathcal P(X)\land\varnothing\notin P\land\forall x\in X\,\exists y\in P\,\forall z\in P\,(x\in z\leftrightarrow y=z)$, but it costs more effort to understand these symbols than to understand "$P$ is a partition of $X$". Feb 3 '20 at 11:36

1. "$$P$$ is a partition of $$X$$" is a concise mathematical way of expressing that $$P$$ is a partition of $$X$$. Edit: There is no standard tidy way to express this in symbols alone; if there were, it would almost certainly be on the Wikipedia page you already reviewed.
2. You could write things like "Choose $$A\in P$$ such that $$x\in A$$.". Something like "$$x$$'s part of the partition" would probably be understood as well. But I think it would be most common to use the relationship between partitions and equivalence relations and just say "the equivalence class of $$x$$" or similar.
3. "a function...the partition it belongs to" in your question is a little unclear. You might have meant just $$f:X\to P$$ where $$f$$ could send $$x\in X$$ somewhere other than its equivalence class. Or if you wanted the function that sends each element of $$X$$ to its equivalence class: that is often written with rectangular brackets, as in "$$x\in [x]$$ and $$[x]\in P$$". Since rectangular brackets can mean other things, you should declare when you're using that notation.