background image

(4)

)

(3).

If

h

S

(

\

(

A

A

))

if

v

g

=

A

for every vertex

v

then

S

(

\

(

A

A

)) =

A

A

. Consider

the remaining case when

V

=

def

h

S

(

\

(

A

A

))

if

v

A

for some vertex

v

. Let

W

=

A

n

V

.

If card

A

= 1

then

S

(

\

(

A

A

))

id

A

=

A

A

; otherwise

W

=

/

;

. Then

V

[

W

=

A

and

so

V

[

]

W

what is equivalent to

V

[

\

(

A

A

)]

W

that is

h

\

(

A

A

)

i

V

\

W

=

/

;

. This is

impossible because

h

\

(

A

A

)

i

V

=

h

\

(

A

A

)

ih

S

(

\

(

A

A

))

i

V

=

h

S

1

(

\

(

A

A

))

i

V

h

S

(

\

(

A

A

))

i

V

=

V

.

(2)

)

(3).

Because

S

(

\

(

A

A

))

A

A

.

Corollary 11.13.

A set

A

is connected regarding a binary relation

i it is connected regarding

\

(

A

A

)

.

Denition 11.14.

A

connected component

of a set

A

regarding a binary relation

F

is a maximal

connected subset of

A

.

Theorem 11.15.

The set

A

is partitioned into connected components (regarding every binary

relation

F

).

Proof.

Consider the binary relation

a

b

,

a

(

S

(

F

))

b

^

b

(

S

(

F

))

a

.

is a symmetric, reexive, and

transitive relation. So all points of

A

are partitioned into a collection of sets

Q

. Obviously each

component is (strongly) connected. If a set

R

A

is greater than one of that connected components

A

then it contains a point

b

2

B

where

B

is some other connected component. Consequently

R

is

disconnected.

Proposition 11.16.

A set is connected (regarding a binary relation) i it has one connected

component.

Proof.

Direct implication is obvious. Reverse is proved by contradiction.

11.4 Connectedness regarding funcoids and reloids

Denition 11.17.

S

1

(

) =

d

f"

RLD

S

1

(

M

)

j

M

2

xyGR

g

for an endoreloid

.

Denition 11.18.

Connectivity reloid

S

(

)

for an endoreloid

is dened as follows:

S

(

) =

l

f"

RLD

S

(

M

)

j

M

2

xyGR

g

:

Remark 11.19.

Do not mess the word

connectivity

with the word

connectedness

which means

being connected.

11.1

Proposition 11.20.

S

(

) =

id

RLD

(

Ob

)

t

S

1

(

)

for every endoreloid

.

Proof.

By the proposition

4.190

.

Proposition 11.21.

S

(

) =

S

(

)

if

is a principal reloid.

Proof.

S

(

) =

d

f

S

(

)

g

=

S

(

)

.

Denition 11.22.

A lter

A 2

F

(

Ob

)

is called

connected

regarding an endoreloid

when

S

(

u

(

RLD

A

))

w A 

RLD

A

.

Obvious 11.23.

A lter

A 2

F

(

Ob

)

is connected regarding an endoreloid

i

S

(

u

(

RLD

A

)) =

RLD

A

.

11.1

In some math literature these two words are used interchangeably.

11.4 Connectedness regarding funcoids and reloids

149