J u m p t o c o n t e n t
M a i n m e n u
M a i n m e n u
N a v i g a t i o n
● M a i n p a g e
● C o n t e n t s
● C u r r e n t e v e n t s
● R a n d o m a r t i c l e
● A b o u t W i k i p e d i a
● C o n t a c t u s
● D o n a t e
C o n t r i b u t e
● H e l p
● L e a r n t o e d i t
● C o m m u n i t y p o r t a l
● R e c e n t c h a n g e s
● U p l o a d f i l e
S e a r c h
Search
A p p e a r a n c e
● C r e a t e a c c o u n t
● L o g i n
P e r s o n a l t o o l s
● C r e a t e a c c o u n t
● L o g i n
P a g e s f o r l o g g e d o u t e d i t o r s l e a r n m o r e
● C o n t r i b u t i o n s
● T a l k
( T o p )
1
D e f i n i t i o n
2
E x a m p l e
3
S e e a l s o
4
R e f e r e n c e s
T o g g l e t h e t a b l e o f c o n t e n t s
R e f l e x i v e c l o s u r e
9 l a n g u a g e s
● D e u t s c h
● Ε λ λ η ν ι κ ά
● E s p a ñ o l
● E u s k a r a
● P o r t u g u ê s
● த ம ி ழ ்
● У к р а ї н с ь к а
● T i ế n g V i ệ t
● 中 文
E d i t l i n k s
● A r t i c l e
● T a l k
E n g l i s h
● R e a d
● E d i t
● V i e w h i s t o r y
T o o l s
T o o l s
A c t i o n s
● R e a d
● E d i t
● V i e w h i s t o r y
G e n e r a l
● W h a t l i n k s h e r e
● R e l a t e d c h a n g e s
● U p l o a d f i l e
● S p e c i a l p a g e s
● P e r m a n e n t l i n k
● P a g e i n f o r m a t i o n
● C i t e t h i s p a g e
● G e t s h o r t e n e d U R L
● D o w n l o a d Q R c o d e
● W i k i d a t a i t e m
P r i n t / e x p o r t
● D o w n l o a d a s P D F
● P r i n t a b l e v e r s i o n
A p p e a r a n c e
F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
Definition [ edit ]
The reflexive closure
S
{\displaystyle S}
of a relation
R
{\displaystyle R}
on a set
X
{\displaystyle X}
is given by
S
=
R
∪
{
(
x
,
x
)
:
x
∈
X
}
{\displaystyle S=R\cup \{(x,x):x\in X\}}
In plain English, the reflexive closure of
R
{\displaystyle R}
is the union of
R
{\displaystyle R}
with the identity relation on
X
.
{\displaystyle X.}
Example [ edit ]
As an example, if
X
=
{
1
,
2
,
3
,
4
}
{\displaystyle X=\{1,2,3,4\}}
R
=
{
(
1
,
1
)
,
(
2
,
2
)
,
(
3
,
3
)
,
(
4
,
4
)
}
{\displaystyle R=\{(1,1),(2,2),(3,3),(4,4)\}}
then the relation
R
{\displaystyle R}
is already reflexive by itself, so it does not differ from its reflexive closure.
However, if any of the pairs in
R
{\displaystyle R}
was absent, it would be inserted for the reflexive closure.
For example, if on the same set
X
{\displaystyle X}
R
=
{
(
1
,
1
)
,
(
2
,
2
)
,
(
4
,
4
)
}
{\displaystyle R=\{(1,1),(2,2),(4,4)\}}
then the reflexive closure is
S
=
R
∪
{
(
x
,
x
)
:
x
∈
X
}
=
{
(
1
,
1
)
,
(
2
,
2
)
,
(
3
,
3
)
,
(
4
,
4
)
}
.
{\displaystyle S=R\cup \{(x,x):x\in X\}=\{(1,1),(2,2),(3,3),(4,4)\}.}
See also [ edit ]
Symmetric closure – operation on binary relationsPages displaying wikidata descriptions as a fallback
Transitive closure – Smallest transitive relation containing a given binary relation
References [ edit ]
t
e
Key concepts
Results
Properties & Types (list )
Constructions
Topology & Orders
Related
t
e
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Reflexive_closure&oldid=1168736844 "
C a t e g o r i e s :
● B i n a r y r e l a t i o n s
● C l o s u r e o p e r a t o r s
● R e w r i t i n g s y s t e m s
● P r o g r a m m i n g l a n g u a g e t h e o r y s t u b s
H i d d e n c a t e g o r i e s :
● P a g e s d i s p l a y i n g w i k i d a t a d e s c r i p t i o n s a s a f a l l b a c k v i a M o d u l e : A n n o t a t e d l i n k
● A l l s t u b a r t i c l 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 4 A u g u s t 2 0 2 3 , a t 1 7 : 4 2 ( U T C ) .
● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n .
● 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
● M o b i l e v i e w