The quotient of
8.5.2 Quotients of Sets by Equivalence Relations
Let
The reason we define quotient sets for equivalence relations only is that each of the properties of being an equivalence relation—reflexivity, symmetry, and transitivity—ensures that the equivalences classes
- Reflexivity. If
is reflexive, then, for each , we have . - Symmetry. The equivalence class
of an element of is defined bybut we could equally well define
instead. This is not a problem when
is symmetric, as we then have .1 - Transitivity. If
is transitive, then and are disjoint iff , and equal otherwise.
Let
- 1.
As a Coequaliser. We have an isomorphism of sets
where
is the equivalence relation generated by . - 2.
As a Pushout. We have an isomorphism of sets1 where
is the equivalence relation generated by . - 3.
The First Isomorphism Theorem for Sets. We have an isomorphism of sets23
- 4.
Descending Functions to Quotient Sets, I. Let
be an equivalence relation on . The following conditions are equivalent:- (a)
There exists a map
making the diagram
commute.
- (b)
We have
. - (c)
For each
, if , then .
- (a)
There exists a map
- 5.
Descending Functions to Quotient Sets, II. Let
be an equivalence relation on . If the conditions of Item 4 hold, then is the unique map making the diagramcommute.
- 6.
Descending Functions to Quotient Sets, III. Let
be an equivalence relation on . We have a bijectionnatural in
, given by the assignment of Item 4 and Item 5, where is the set defined by - 7.
Descending Functions to Quotient Sets, IV. Let
be an equivalence relation on . If the conditions of Item 4 hold, then the following conditions are equivalent:- (a)
The map
is an injection. - (b)
We have
. - (c)
For each
, we have iff .
- (a)
The map
- 8.
Descending Functions to Quotient Sets, V. Let
be an equivalence relation on . If the conditions of Item 4 hold, then the following conditions are equivalent:- (a)
The map
is surjective. - (b)
The map
is surjective.
- (a)
The map
- 9.
Descending Functions to Quotient Sets, VI. Let
be a relation on and let be the equivalence relation associated to . The following conditions are equivalent:
of
- There exist
satisfying at least one of the following conditions:- (a)
The following conditions are satisfied:
- (i)
We have
or ; - (ii)
We have
or for each ; - (iii)
We have
or ;
- (i)
We have
- (b)
We have
.
- (a)
The following conditions are satisfied:
and