In mathematics, a binary relation R ⊆ X×Y between two sets X and Y is total (or left total) if the source set X equals the domain {x : there is a y with xRy }. Conversely, R is called right total if Y equals the range {y : there is an x with xRy }.
When f: X → Y is a function, the domain of f is all of X, hence f is a total relation. On the other hand, if f is a partial function, then the domain may be a proper subset of X, in which case f is not a total relation.
"A binary relation is said to be total with respect to a universe of discourse just in case everything in that universe of discourse stands in that relation to something else."[1]
Total relations can be characterized algebraically by equalities and inequalities involving compositions of relations. To this end, let
be two sets, and let
For any two sets
let
be the universal relation between
and
and let
be the identity relation on
We use the notation
for the converse relation of ![{\displaystyle R.}](//wikimedia.org/api/rest_v1/media/math/render/svg/fdcae6b33a27f86c7961318cd7ee3d789d3bcdd2)
is total iff for any set
and any
implies
[2]: 54
is total iff
[2]: 54
- If
is total, then
The converse is true if
[note 1]
- If
is total, then
The converse is true if
[note 2][2]: 63
- If
is total, then
The converse is true if
[2]: 54 [3]
- More generally, if
is total, then for any set
and any
The converse is true if
[note 3][2]: 57
If
then
will be not total.
Observe
and apply the previous bullet.
Take
and appeal to the previous bullet.