• Source: Total relation

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."




Algebraic characterization


Total relations can be characterized algebraically by equalities and inequalities involving compositions of relations. To this end, let



X
,
Y


{\displaystyle X,Y}

be two sets, and let



R
⊆
X
×
Y
.


{\displaystyle R\subseteq X\times Y.}

For any two sets



A
,
B
,


{\displaystyle A,B,}

let




L

A
,
B


=
A
×
B


{\displaystyle L_{A,B}=A\times B}

be the universal relation between



A


{\displaystyle A}

and



B
,


{\displaystyle B,}

and let




I

A


=
{
(
a
,
a
)
:
a
∈
A
}


{\displaystyle I_{A}=\{(a,a):a\in A\}}

be the identity relation on



A
.


{\displaystyle A.}

We use the notation




R

⊤




{\displaystyle R^{\top }}

for the converse relation of



R
.


{\displaystyle R.}





R


{\displaystyle R}

is total iff for any set



W


{\displaystyle W}

and any



S
⊆
W
×
X
,


{\displaystyle S\subseteq W\times X,}





S
≠
∅


{\displaystyle S\neq \emptyset }

implies



S
R
≠
∅
.


{\displaystyle SR\neq \emptyset .}

: 54 




R


{\displaystyle R}

is total iff




I

X


⊆
R

R

⊤


.


{\displaystyle I_{X}\subseteq RR^{\top }.}

: 54 
If



R


{\displaystyle R}

is total, then




L

X
,
Y


=
R

L

Y
,
Y


.


{\displaystyle L_{X,Y}=RL_{Y,Y}.}

The converse is true if



Y
≠
∅
.


{\displaystyle Y\neq \emptyset .}


If



R


{\displaystyle R}

is total, then






R

L

Y
,
Y



¯


=
∅
.


{\displaystyle {\overline {RL_{Y,Y}}}=\emptyset .}

The converse is true if



Y
≠
∅
.


{\displaystyle Y\neq \emptyset .}

: 63 
If



R


{\displaystyle R}

is total, then





R
¯


⊆
R



I

Y


¯


.


{\displaystyle {\overline {R}}\subseteq R{\overline {I_{Y}}}.}

The converse is true if



Y
≠
∅
.


{\displaystyle Y\neq \emptyset .}

: 54 
More generally, if



R


{\displaystyle R}

is total, then for any set



Z


{\displaystyle Z}

and any



S
⊆
Y
×
Z
,


{\displaystyle S\subseteq Y\times Z,}








R
S

¯


⊆
R


S
¯


.


{\displaystyle {\overline {RS}}\subseteq R{\overline {S}}.}

The converse is true if



Y
≠
∅
.


{\displaystyle Y\neq \emptyset .}

: 57 




See also


Serial relation — a total homogeneous relation




Notes






References



Gunther Schmidt & Michael Winter (2018) Relational Topology
C. Brink, W. Kahl, and G. Schmidt (1997) Relational Methods in Computer Science, Advances in Computer Science, page 5, ISBN 3-211-82971-7
Gunther Schmidt & Thomas Strohlein (2012)[1987] Relations and Graphs, p. 54, at Google Books
Gunther Schmidt (2011) Relational Mathematics, p. 57, at Google Books

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