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( R e d i r e c t e d f r o m L e f t - t o t a l )
In mathematics , a binary relation R X Y X Y is total (or left total ) if the source set X x y xRy }. Conversely, R right total if Y y x xRy }.
When f X Y function , the domain of f X f f partial function , then the domain may be a proper subset of X f
"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 ]
Algebraic characterization
edit
Total relations can be characterized algebraically by equalities and inequalities involving compositions of relations . To this end, let
X ,
Y
{\displaystyle X,Y}
R ⊆
X ×
Y .
{\displaystyle R\subseteq X\times Y.}
A ,
B ,
{\displaystyle A,B,}
L
A ,
B
=
A ×
B
{\displaystyle L_{A,B}=A\times B}
universal relation between
A
{\displaystyle A}
B ,
{\displaystyle B,}
I
A
=
{
(
a ,
a )
:
a ∈
A }
{\displaystyle I_{A}=\{(a,a):a\in A\}}
identity relation on
A .
{\displaystyle A.}
R
⊤
{\displaystyle R^{\top }}
converse relation of
R .
{\displaystyle R.}
R
{\displaystyle R}
W
{\displaystyle W}
S ⊆
W ×
X ,
{\displaystyle S\subseteq W\times X,}
S ≠
∅
{\displaystyle S\neq \emptyset }
S R ≠
∅
.
{\displaystyle SR\neq \emptyset .}
[2 ] : 54
R
{\displaystyle R}
I
X
⊆
R
R
⊤
.
{\displaystyle I_{X}\subseteq RR^{\top }.}
[2 ] : 54
If
R
{\displaystyle R}
L
X ,
Y
=
R
L
Y ,
Y
.
{\displaystyle L_{X,Y}=RL_{Y,Y}.}
Y ≠
∅
.
{\displaystyle Y\neq \emptyset .}
[note 1]
If
R
{\displaystyle R}
R
L
Y ,
Y
¯
=
∅
.
{\displaystyle {\overline {RL_{Y,Y}}}=\emptyset .}
Y ≠
∅
.
{\displaystyle Y\neq \emptyset .}
[note 2] [2 ] : 63
If
R
{\displaystyle R}
R ¯
⊆
R
I
Y
¯
.
{\displaystyle {\overline {R}}\subseteq R{\overline {I_{Y}}}.}
Y ≠
∅
.
{\displaystyle Y\neq \emptyset .}
[2 ] : 54 [3 ]
More generally, if
R
{\displaystyle R}
Z
{\displaystyle Z}
S ⊆
Y ×
Z ,
{\displaystyle S\subseteq Y\times Z,}
R S
¯
⊆
R
S ¯
.
{\displaystyle {\overline {RS}}\subseteq R{\overline {S}}.}
Y ≠
∅
.
{\displaystyle Y\neq \emptyset .}
[note 3] [2 ] : 57
See also
edit
Notes
edit
^ If
Y =
∅
≠
X ,
{\displaystyle Y=\emptyset \neq X,}
R
{\displaystyle R}
^ Observe
R
L
Y ,
Y
¯
=
∅
⇔
R
L
Y ,
Y
=
L
X ,
Y
,
{\displaystyle {\overline {RL_{Y,Y}}}=\emptyset \Leftrightarrow RL_{Y,Y}=L_{X,Y},}
^ Take
Z =
Y ,
S =
I
Y
{\displaystyle Z=Y,S=I_{Y}}
References
edit
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Total_relation&oldid=1204646531 "
L a s t e d i t e d o n 7 F e b r u a r y 2 0 2 4 , a t 1 5 : 3 0 L a n g u a g e s
T h i s p a g e i s n o t a v a i l a b l e i n o t h e r l a n g u a g e s .
● T h i s p a g e w a s l a s t e d i t e d o n 7 F e b r u a r y 2 0 2 4 , a t 1 5 : 3 0 ( U T C ) . ● C o n t e n t i s a v a i l a b l e u n d e r C C B Y - S A 4 . 0 u n l e s s o t h e r w i s e n o t e d . ● P r i v a c y p o l i c y ● A b o u t W i k i p e d i a ● D i s c l a i m e r s ● C o n t a c t W i k i p e d i a ● C o d e o f C o n d u c t ● D e v e l o p e r s ● S t a t i s t i c s ● C o o k i e s t a t e m e n t ● T e r m s o f U s e ● D e s k t o p
is total iff for any set 
and any 
implies 
[2]: 54 
[2]: 54 
The converse is true if 
[note 1]
The converse is true if 
The converse is true if 
and any 
The converse is true if 
and appeal to the previous bullet.
