Geoscience Reference
In-Depth Information
X
2
X
1
X
Par
f
1
,f
2
encl
X
gen
Par
f
1
,f
2
,f
3
X
Par
f
1
,f
3
X
Par
f
2
,f
3
X
3
Par
f
1
;f
2
;f
3
gen
Fig. 9.11
The enclosure of
X
Lemma 9.2
The following inclusion of sets holds:
cl
encl
X
Par
f
1
;f
2
;f
3
X
sPar
f
1
;f
2
;f
3
:
gen
The interested reader is referred to Nickel et al. (
2005b
) for a detailed proof of this
result.
X
Par
f
i
;f
j
D
X
wPar
f
i
;f
j
for any i;j
2f
1;2;3
g
,we
Remark 9.4
Since
have that:
encl
X
Par
f
1
;f
2
;f
3
D
encl
X
wPar
f
1
;f
2
;f
3
:
gen
gen
Finally we obtain the following theorem which provides a subset as well as a
superset of
X
Par
f
1
;f
2
;f
3
.
Theorem 9.5
The following inclusions of sets hold:
encl
X
Par
f
1
;f
2
;f
3
X
Par
f
1
;f
2
;f
3
gen
Par
f
1
;f
2
;f
3
[
encl
X
Par
f
1
;f
2
;f
3
gen
gen
X
wPar
f
1
;f
2
;f
3
:
D
X
Proof
Using Lemma
9.2
and Theorem
9.2
we have the following chain of inclusions
that proves the thesis of the theorem.
encl
X
Par
f
1
;f
2
;f
3
sPar
f
1
;f
2
;f
3
gen
X
Par
f
1
;f
2
;f
3
wPar
f
1
;f
2
;f
3
X
X
Par
f
1
;f
2
;f
3
[
encl
X
Par
f
1
;f
2
;f
3
:
gen
gen
X
