# The meaning of $Z_a$, where $Z$ is a partitioning of $A$ and $a \in A$

I am unsure about the common usage of subscripting a set with something, more precisely something which might be a member of that set or some other set (as opposed to, say, subscripting a set with an integer for reasons of indexing).

The text where I encountered this usage, for context:

Let $Z$ be a partitioning. Define the binary relation $\equiv_Z \subseteq A \times A$ by, for all $a_1,a_2 \in A$, $a_1 \equiv_Z a_2$ iff $Z_{a_1} = Z_{a_2}$.

($Z$ is presumably a partitioning of $A$.)

$\equiv_Z$ is an equivalence relation, as can be shown (it is one of the exercises).

I interpret this to mean that $Z_{a_1}$ is a set, $Z_{a_1} \in Z$ and that $a_1 \in Z_{a_1}$, i.e. that $Z_{a_1}$ is the set in $Z$ that contains $a_1$ from $A$. Is this correct, or in line with most conventional notation?

• Perhaps saying where you seen this might also help potential answerers. (Name of the book, if it is from some textbook. Link if it is from some lecture notes available online.) Nov 26, 2013 at 13:41
• @bof This is done more often, and there is a distinction: ProofWiki: Partitioning, ProofWiki: Partition. Nov 26, 2013 at 13:47
• @MartinSleziak it is from an exercise about set theory (sets, functions, equivalence relations, quotient sets, Barber's Paradox...). The course is about using set theory to define data types and operations on these data types (signatures), and their meaning (given by algebras), but this exercise set is only about set theory. Nov 26, 2013 at 13:51
• @bof the quotation is not a translation, and it does say "partitioning". Nov 26, 2013 at 13:53

This, i.e. $Z_a$ being defined as the unique set with $Z_a \in Z: a \in Z_a$, seems to be the only sensible option.
More so since $\equiv_Z$ is a symbol that looks like it's supposed to be an equivalence relation (which it is, given this definition).
In cases like these, the subscript effectively acts as an argument to a function. That is, we can think of $Z_a$ as an alternative notation to $Z(a)$.
I'm not sure that this notation is conventional, but then, most sources I have seen go the other way around, deriving the partitioning $Z$ from a given equivalence relation $\equiv_Z$.