Notation for partitions of sets

Question

Given the wikipedia definition of partition of a set:

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.

Answer 1

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.