quotient set

Quotient sets


Given a set SS and an equivalence relation \equiv on SS, the quotient set of SS by \equiv is the set S/S/{\equiv} whose elements are the elements of SS but where two elements are now considered equal if in SS they were merely equivalent.

If changing the definition of equality like this is not allowed in your foundations of mathematics, then you can still define S/S/{\equiv} as a subset of the power set of SS as follows:

(1)AS/(x:S),A={y:Sxy} A \in S/{\equiv} \;\Leftrightarrow\; \exists (x: S),\; A = \{y: S \;\mid\; x \equiv y\}

If you don't have (extensional) power sets either, then you'll have to use setoids (in which case, perhaps you'd do better to change your terminology as described there). Alternatively, don't worry about any of this and just include the existence of quotient objects in Set as an axiom (the axiom of quotient sets, which you have if you assume that Set\Set is a pretopos or a Grothendieck topos as given by Giraud's axioms).

In any case, the element of S/S/{\equiv} that comes from the element xx of SS may be denoted [x] [x]_{\equiv}, or simply [x][x] if {\equiv} is understood, or simply xx if there will be no confusion as to which set it is an element of. This [x][x] is called the equivalence class of xx with respect to \equiv; the term ‘class’ here is an old word for ‘set’ (in the sense of ‘subset’) and refers to the definition (1) above, where [x][x] is literally the set AA.


Quotient sets in Set generalise to quotient objects in other categories. In particular, an exact category is a regular category in which every congruence on every object has an effective quotient object.

Last revised on September 3, 2012 at 13:40:08. See the history of this page for a list of all contributions to it.