[set theory] relational closure (reflexive closure | symmetric closure | transitive closure)

One 、 Relational closure

---

Contains the given element , also Have specified properties Of The smallest aggregate , It is called the closure of relation ; The nature of this designation is relation

R

R

R

Reflexive closure r ( R ) : contain

R

R

R Relationship , towards

R

R

R In relationship , Add ordered pairs , become introspect Of The smallest binary relation

Symmetric closure s ( R ) : contain

R

R

R Relationship , towards

R

R

R In relationship , Add ordered pairs , become symmetry Of The smallest binary relation

Pass closures t ( R ) : contain

R

R

R Relationship , towards

R

R

R In relationship , Add ordered pairs , Turn into transmission Of The smallest binary relation

There are three important elements in the definition :

• Contains the given element
• Have specified properties
• The smallest binary relation

Two 、 Reflexive closure

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Reflexive closure r ( R ) : contain

R

R

R Relationship , towards

R

R

R In relationship , Add ordered pairs , become introspect Of The smallest binary relation

R

⊆

r

(

R

)

R \subseteq r(R)

R⊆r(R)

r

(

R

)

r(R)

r(R) It's reflexive

∀

S

(

(

R

⊆

S

∧

S

since

back

)

→

r

(

R

)

⊆

S

)

\forall S ( ( R \subseteq S\land S introspect ) \to r(R) \subseteq S)

∀S((R⊆S∧S since back )→r(R)⊆S)

Relationship

R

R

R Diagram for

G

(

R

)

G(R)

G(R) :

R

R

R Reflexive closure of

G

(

r

(

R

)

)

G(r ( R ))

G(r(R)) The diagram : stay

R

R

R On the basis of , Add some ordered pairs , send

r

(

R

)

r(R)

r(R) become introspect Of The smallest binary relation , The reflexive condition is that all vertices have rings , Here we add rings for all four vertices ;

3、 ... and 、 Symmetric closure

---

Reflexive closure r ( R ) : contain

R

R

R Relationship , towards

R

R

R In relationship , Add ordered pairs , become symmetry Of The smallest binary relation

R

⊆

s

(

R

)

R \subseteq s(R)

R⊆s(R)

s

(

R

)

s(R)

s(R) It's symmetrical

∀

S

(

(

R

⊆

S

∧

S

Yes

call

)

→

r

(

R

)

⊆

S

)

\forall S ( ( R \subseteq S\land S symmetry ) \to r(R) \subseteq S)

∀S((R⊆S∧S Yes call )→r(R)⊆S)

Relationship

R

R

R Diagram for

G

(

R

)

G(R)

G(R) :

R

R

R Symmetric closure of

G

(

s

(

R

)

)

G(s ( R ))

G(s(R)) The diagram : stay

R

R

R On the basis of , Add some ordered pairs , send

s

(

R

)

s(R)

s(R) become symmetry Of The smallest binary relation , The condition of symmetry is Between any two vertices

0

/

2

0/2

0/2 The strip has a directed edge , Yes

0

0

0 Regardless of the edge , Yes

1

1

1 Add a reverse directed edge ;

Four 、 Pass closures

---

Reflexive closure r ( R ) : contain

R

R

R Relationship , towards

R

R

R In relationship , Add ordered pairs , become Pass on Of The smallest binary relation

R

⊆

t

(

R

)

R \subseteq t(R)

R⊆t(R)

t

(

R

)

t(R)

t(R) It's symmetrical

∀

S

(

(

R

⊆

S

∧

S

Pass on

Deliver

)

→

r

(

R

)

⊆

S

)

\forall S ( ( R \subseteq S\land S Pass on ) \to r(R) \subseteq S)

∀S((R⊆S∧S Pass on Deliver )→r(R)⊆S)

Relationship

R

R

R Diagram for

G

(

R

)

G(R)

G(R) :

R

R

R Symmetric closure of

G

(

t

(

R

)

)

G(t ( R ))

G(t(R)) The diagram : stay

R

R

R On the basis of , Add some ordered pairs , send

t

(

R

)

t(R)

t(R) become Pass on Of The smallest binary relation , The condition of transmission is ① Premise

a

→

b

,

b

→

c

a\to b, b \to c

a→b,b→c establish ,

a

→

c

a \to c

a→c There is , or ② The premise doesn't hold , If the premise is not tenable, it is transmitted by default , If the premise holds , You must add the corresponding third side ;