[set theory] relation properties (reflexivity | reflexivity theorem | reflexivity | reflexivity theorem | example)

One 、 reflexivity

---

Reflexive symbolic description :

R

⊆

A

×

A

R \subseteq A \times A

R⊆A×A

R

R

R The relationship is Reflexive

⇔

\Leftrightarrow

⇔

∀

x

(

x

∈

A

→

x

R

x

)

\forall x ( x \in A \to xRx )

∀x(x∈A→xRx)

⇔

\Leftrightarrow

⇔

(

∀

x

∈

A

)

x

R

x

(\forall x \in A) xRx

(∀x∈A)xRx

Non reflexive symbolic description :

R

R

R Right and wrong reflexive

⇔

\Leftrightarrow

⇔

∃

x

(

x

∈

A

∧

¬

x

R

x

)

\exist x( x \in A \land \lnot xRx )

∃x(x∈A∧¬xRx)

Reflexive textual description :

R

R

R yes

A

A

A Binary relations on sets ,

R

R

R It's reflexive ,

If and only if

R

R

R In the collection , arbitrarily

x

x

x Belong to a collection

A

A

A The elements of ,

x

x

x And

x

x

x Have a relationship

R

R

R( Must be all

x

x

x )

irreflexive A word description : There is

x

x

x Elements ,

x

x

x Belong to

A

A

A The elements in the collection , also

x

x

x And

x

x

x It doesn't matter. ;

reflexivity It's validation Every element Rather than itself There are

R

R

R Relationship

Non reflexivity As long as there is one element Rather than itself No,

R

R

R The relationship is established

∅

\varnothing

∅ Empty relation on , Is reflexive , It is reflexive again

Two 、 Reflexivity theorem

---

Reflexivity theorem :

R

R

R It's reflexive

⇔

\Leftrightarrow

⇔

I

A

⊆

R

I_A \subseteq R

IA​⊆R

⇔

\Leftrightarrow

⇔

R

−

1

yes

since

back

Of

R^{-1} It's reflexive

R−1 yes since back Of

⇔

\Leftrightarrow

⇔

M

(

R

)

M(R)

M(R) The values on the main diagonal of the relationship matrix are

1

1

1

⇔

\Leftrightarrow

⇔

G

(

R

)

G(R)

G(R) Every vertex in the graph has a ring

A word description :

R

R

R It's reflexive

If and only if

R

R

R Including identity relation ,

I

A

⊆

R

I_A \subseteq R

IA​⊆R

If and only if

R

−

1

R^{-1}

R−1 It's reflexive

If and only if

M

(

R

)

M(R)

M(R) The elements on the main diagonal of the relation matrix are all

1

1

1

If and only if

G

(

R

)

G(R)

G(R) Every vertex in the graph has a ring

3、 ... and 、 Reflexivity

---

Reflexivity :

R

⊆

A

×

A

R \subseteq A \times A

R⊆A×A

R

R

R It's anti reflexive

⇔

\Leftrightarrow

⇔

∀

x

(

x

∈

A

→

¬

x

R

x

)

\forall x ( x \in A \to \lnot xRx )

∀x(x∈A→¬xRx)

⇔

\Leftrightarrow

⇔

(

∀

x

∈

A

)

¬

x

R

x

(\forall x \in A) \lnot xRx

(∀x∈A)¬xRx

The diagram :

introspect Every point They all have rings ( a key )

irreflexive yes Some have rings , Some have no rings

Reflexion Every point There is no ring ( a key )

Non reflexive yes Some have rings , Some have no rings

∅

\varnothing

∅ Empty relation on , Is reflexive , It is reflexive again

Four 、 Anti reflexivity theorem

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Reflexive theorem :

R

R

R It's anti reflexive

⇔

\Leftrightarrow

⇔

I

A

∩

R

=

∅

I_A \cap R = \varnothing

IA​∩R=∅

⇔

\Leftrightarrow

⇔

R

−

1

R^{-1}

R−1 It's anti reflexive

⇔

\Leftrightarrow

⇔

M

(

R

)

M(R)

M(R) The elements on the main diagonal are

0

0

0

⇔

\Leftrightarrow

⇔

G

(

R

)

G(R)

G(R) There is no ring at every vertex

A word description :

R

R

R It's anti reflexive

If and only if Relationship

R

R

R And Identity

I

A

I_A

IA​ Disjoint

If and only if The inverse of the relationship

R

−

1

R^{-1}

R−1 It's anti reflexive

If and only if The relational matrix

M

(

R

)

M(R)

M(R) The elements on the main diagonal are all

0

0

0

If and only if The diagram

G

(

R

)

G(R)

G(R) Every vertex of has no ring

5、 ... and 、 Examples of reflexivity and reflexivity

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In the above diagram , Every vertex has a ring , It's reflexive ;

In the above diagram , Every vertex has no ring , It's anti reflexive

In the above diagram , Some vertices have rings , Some vertices have no rings , Nothing ;