An Extended Relational Model & SQL for Fuzzy Multidatabases - Advanced Database Query Systems - page 211

Lemma 1 5 . ∀ L , ..., L L , , ..., L and G , = merge L ( , ..., L ),

1 j nj 1 k nk j 1 j nj

fs fs

fs

G merge L ( , ..., L G ),

sameDB

G merge L ( L , ..., L L ).

= =

DD DD

DD

k 1 k nk j p α β , k 1 j p α β , 1 k nj p α β , nk

fs

sameDB

Proof: ∈ g G ( G ) ∧ g s . = DB for some i ∈ { , ..., } 1 n ⇔ DD

j p α β , k i

(( . g A , µ ( . g A DB ), ) ∈ G ) ∧ (( . g A , µ Gk ( . g A DB ), ) ∈ G ) ∧

j Gj j i j k k i k

)) p g A g A for some i ( . , . )

µ ( ) min( g = µ ( . g A ), µ ( . g A  β ∧ ⇔

f

Gj j Gk k α j k

( Gj Gk ) DD

p α β ,

( . g A ∈ L ∧ µ ( . g A )  β ) ∧ (( . g A ∈ L ∧ µ ( . g A )  β ) ∧ p g A g A for some i ( . , . ) ⇔

j i j , L j k i ik , L k α j k

ij ik

f

(( . g A , µ ( . g A )),( . g A , µ ( . g A ))) ∈ ( L L ) for some i ⇔

DD

j Gj j k L k ij p α β , ik

ik

g = (( . g A , g A . ),min( µ ( . g A ), µ ( . g A )), DB for some i ) ⇔

j k L j L k i

ij ik

f f

merge L ( L , ..., L L ).

1 DD

DD

j p α β , 1 k nj p α β , nk

Lemma 1 6 . ∀ L , ..., L L , , ..., L and G , = merge L ( , ..., L ),

1 j nj 1 k nk j 1 j nj

fs fs

fs

sameDB

G = merge L ( , ..., L G ), G = merge L ( L , ..., L L ).

∪ ∪ ∪

k 1 k nk j p α β , k 1 j p α β , 1 k nj p α β , nk

fs

fs

sameDB

Proof: Let G = ( G G and Le ) t L = ( L L ).

∪ ∪

j p α β , k ij α β ik

g G g s DB for some i ∈ ∧ . = ∈ { , ..., } 1 n ⇔

i

fs

sameDB

g G ∈ ( G ) ∧ g s DB for some i . = ∈ { , ..., } 1 n ⇔

∪

j p α β , k i

( ( . g A g A , . ) ( . ) max( g A ( . ), g A ( g A . )) ) ( ( . g A g A , . )

µ  α ∧ µ = µ µ  β ∨ µ 

EQ Gj G G Gj Gj EQ Gk

α µ ∧ ( g . ) max( A = µ ( . ), g A µ ( g A . ))  β ) ∧ g s DB for some i . = ∈ { , ..., } 1 n ⇔

G G Gk Gk i

( µ ( . g A g A , . )  α µ ∧ ( . ) max( g A = µ ( . ), g A µ ( g A . ))  β ) ∨ µ ( ( . g A L A , . ) 

EQ L L L L Lij EQ ik

ij ij

α µ ∧ ( . ) max( g A = µ ( . ), g A µ ( g A . ))  β ) for s ome i ∈ { , ..., } 1 n ⇔

L L L L

ik ik

f

g A . ( L L ) for some i { , ..., } 1 n

∈ ∈ ⇔

∪

ij α β , ik

f f

merge ( L L , ..., L L ). ∪ ∪

ij α β , ik nj α β , nk

Lemma 1 7 . ∀ L , ..., L L , , ..., L and G , = merge L ( , ..., L ),

1 j nj 1 k nk j 1 j nj

f f

fs sameDB

G = merge L ( , ..., L G ), G = merge L ( L , ..., L L ).

∩ ∩ ∩

k 1 k nk j α β , k 1 j α β , 1 k nj α β , nk

f

fs sameDB

Proof: Let G = ( G G and Let L ) = ( L L ).

∩ ∩

j α β , k i j α β , ik

g G g s DB for some i ∈ ∧ . = ∈ { , ..., } 1 n ⇔

i

fs sameD

B

g G ∈ ( G ) ∧ g s DB for some i . = ∈ { , ..., } 1 n ⇔

∩

j α β , k i

( µ ( . g A g A , . )  α µ ∧ ( . ) g A = max( µ ( . ), g A µ ( g A . ))  β ) ∨ ( µ ( . g A g A , . ) 

EQ Gj G G Gj Gj EQ Gk

α µ ∧ ( . ) max( g A = µ ( . ), g A µ ( g A . ))  β ) ∧ g s DB for some i . = ∈ { , ..., } 1 n ⇔

G G Gk Gk i

( µ ( . g A , g A . )  α µ ∧ ( . ) max( g A = µ ( . ), g A µ ( g A . ))  β ) ∨ ( µ ( . g A , L A . ) 

EQ L L L L Lij EQ ik

ij ij

α µ ∧ ( . ) max( g A = µ ( . ), g A µ ( g A . ))  β ) for some i ∈ 1 ..., } { , n ⇔

L L L L

ik ik

f

g A . ∈ ( L L ) for some i ∈ { , ..., } 1 n ⇔

∩

ij α β , ik

f f

merge L ( L , ..., L L ).

∩ ∩ α β

ij α β , ik nj , nk